Jmat.Real.erf

Percentage Accurate: 78.8% → 99.0%

Time: 30.8s

Alternatives: 16

Speedup: 8.3×

Specification

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\\ 1 - \left(t_0 \cdot \left(0.254829592 + t_0 \cdot \left(-0.284496736 + t_0 \cdot \left(1.421413741 + t_0 \cdot \left(-1.453152027 + t_0 \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))))
   (-
    1.0
    (*
     (*
      t_0
      (+
       0.254829592
       (*
        t_0
        (+
         -0.284496736
         (*
          t_0
          (+ 1.421413741 (* t_0 (+ -1.453152027 (* t_0 1.061405429)))))))))
     (exp (- (* (fabs x) (fabs x))))))))

C

double code(double x) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * fabs(x)));
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * exp(-(fabs(x) * fabs(x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = 1.0d0 / (1.0d0 + (0.3275911d0 * abs(x)))
    code = 1.0d0 - ((t_0 * (0.254829592d0 + (t_0 * ((-0.284496736d0) + (t_0 * (1.421413741d0 + (t_0 * ((-1.453152027d0) + (t_0 * 1.061405429d0))))))))) * exp(-(abs(x) * abs(x))))
end function

Java

public static double code(double x) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * Math.abs(x)));
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * Math.exp(-(Math.abs(x) * Math.abs(x))));
}

Python

def code(x):
	t_0 = 1.0 / (1.0 + (0.3275911 * math.fabs(x)))
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * math.exp(-(math.fabs(x) * math.fabs(x))))

Julia

function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x))))
	return Float64(1.0 - Float64(Float64(t_0 * Float64(0.254829592 + Float64(t_0 * Float64(-0.284496736 + Float64(t_0 * Float64(1.421413741 + Float64(t_0 * Float64(-1.453152027 + Float64(t_0 * 1.061405429))))))))) * exp(Float64(-Float64(abs(x) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / (1.0 + (0.3275911 * abs(x)));
	tmp = 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * exp(-(abs(x) * abs(x))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[(N[(t$95$0 * N[(0.254829592 + N[(t$95$0 * N[(-0.284496736 + N[(t$95$0 * N[(1.421413741 + N[(t$95$0 * N[(-1.453152027 + N[(t$95$0 * 1.061405429), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Initial Program: 78.8% accurate, 1.0× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + 0.3275911 \cdot \left|x\right|}\\ 1 - \left(t_0 \cdot \left(0.254829592 + t_0 \cdot \left(-0.284496736 + t_0 \cdot \left(1.421413741 + t_0 \cdot \left(-1.453152027 + t_0 \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x))))))
   (-
    1.0
    (*
     (*
      t_0
      (+
       0.254829592
       (*
        t_0
        (+
         -0.284496736
         (*
          t_0
          (+ 1.421413741 (* t_0 (+ -1.453152027 (* t_0 1.061405429)))))))))
     (exp (- (* (fabs x) (fabs x))))))))

C

double code(double x) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * fabs(x)));
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * exp(-(fabs(x) * fabs(x))));
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: t_0
    t_0 = 1.0d0 / (1.0d0 + (0.3275911d0 * abs(x)))
    code = 1.0d0 - ((t_0 * (0.254829592d0 + (t_0 * ((-0.284496736d0) + (t_0 * (1.421413741d0 + (t_0 * ((-1.453152027d0) + (t_0 * 1.061405429d0))))))))) * exp(-(abs(x) * abs(x))))
end function

Java

public static double code(double x) {
	double t_0 = 1.0 / (1.0 + (0.3275911 * Math.abs(x)));
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * Math.exp(-(Math.abs(x) * Math.abs(x))));
}

Python

def code(x):
	t_0 = 1.0 / (1.0 + (0.3275911 * math.fabs(x)))
	return 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * math.exp(-(math.fabs(x) * math.fabs(x))))

Julia

function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(0.3275911 * abs(x))))
	return Float64(1.0 - Float64(Float64(t_0 * Float64(0.254829592 + Float64(t_0 * Float64(-0.284496736 + Float64(t_0 * Float64(1.421413741 + Float64(t_0 * Float64(-1.453152027 + Float64(t_0 * 1.061405429))))))))) * exp(Float64(-Float64(abs(x) * abs(x))))))
end

MATLAB

function tmp = code(x)
	t_0 = 1.0 / (1.0 + (0.3275911 * abs(x)));
	tmp = 1.0 - ((t_0 * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + (t_0 * (-1.453152027 + (t_0 * 1.061405429))))))))) * exp(-(abs(x) * abs(x))));
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(0.3275911 * N[Abs[x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, N[(1.0 - N[(N[(t$95$0 * N[(0.254829592 + N[(t$95$0 * N[(-0.284496736 + N[(t$95$0 * N[(1.421413741 + N[(t$95$0 * N[(-1.453152027 + N[(t$95$0 * 1.061405429), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Exp[(-N[(N[Abs[x], $MachinePrecision] * N[Abs[x], $MachinePrecision]), $MachinePrecision])], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Alternative 1: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, x \cdot \left(x \cdot -1.2734922589667106\right)\right) + \mathsf{fma}\left(x, 2.256772713883663, 1.999999999 \cdot 10^{-9}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.254829592 + \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{t_0}}{t_0}}{t_0}}{t_0}\right)\right)}{t_0 \cdot {\left(e^{x}\right)}^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x) 1.0)))
   (if (<= (fabs x) 5e-12)
     (/
      (+
       (fma (pow x 3.0) -0.7506356576441799 (* x (* x -1.2734922589667106)))
       (fma x 2.256772713883663 1.999999999e-9))
      (+
       1.0
       (/
        (+
         0.254829592
         (/
          (+
           -0.284496736
           (/
            (+
             1.421413741
             (/
              (+ -1.453152027 (/ 1.061405429 (fma x 0.3275911 1.0)))
              (fma x 0.3275911 1.0)))
            (fma x 0.3275911 1.0)))
          (fma x 0.3275911 1.0)))
        (* (fma x 0.3275911 1.0) (exp (* x x))))))
     (-
      1.0
      (/
       (+
        0.254829592
        (expm1
         (log1p
          (/
           (+
            -0.284496736
            (/
             (+ 1.421413741 (/ (+ -1.453152027 (/ 1.061405429 t_0)) t_0))
             t_0))
           t_0))))
       (* t_0 (pow (exp x) x)))))))

C

double code(double x) {
	double t_0 = fma(0.3275911, fabs(x), 1.0);
	double tmp;
	if (fabs(x) <= 5e-12) {
		tmp = (fma(pow(x, 3.0), -0.7506356576441799, (x * (x * -1.2734922589667106))) + fma(x, 2.256772713883663, 1.999999999e-9)) / (1.0 + ((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / (fma(x, 0.3275911, 1.0) * exp((x * x)))));
	} else {
		tmp = 1.0 - ((0.254829592 + expm1(log1p(((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / t_0)) / t_0)) / t_0)) / t_0)))) / (t_0 * pow(exp(x), x)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(0.3275911, abs(x), 1.0)
	tmp = 0.0
	if (abs(x) <= 5e-12)
		tmp = Float64(Float64(fma((x ^ 3.0), -0.7506356576441799, Float64(x * Float64(x * -1.2734922589667106))) + fma(x, 2.256772713883663, 1.999999999e-9)) / Float64(1.0 + Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / Float64(fma(x, 0.3275911, 1.0) * exp(Float64(x * x))))));
	else
		tmp = Float64(1.0 - Float64(Float64(0.254829592 + expm1(log1p(Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / t_0)) / t_0)) / t_0)) / t_0)))) / Float64(t_0 * (exp(x) ^ x))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 5e-12], N[(N[(N[(N[Power[x, 3.0], $MachinePrecision] * -0.7506356576441799 + N[(x * N[(x * -1.2734922589667106), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 2.256772713883663 + 1.999999999e-9), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * 0.3275911 + 1.0), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(0.254829592 + N[(Exp[N[Log[1 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision]), $MachinePrecision] / N[(t$95$0 * N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 4.9999999999999997e-12

   1. Initial program 57.7%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Applied egg-rr 57.7%

      \[\leadsto \color{blue}{\frac{1 - \left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}\right) \cdot \left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}}} \]

   5. Simplified 57.7%

      \[\leadsto \color{blue}{\frac{1 + \frac{\frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{e^{x \cdot x}} \cdot \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}} \]

   6. Taylor expanded in x around 0 99.6%

      \[\leadsto \frac{\color{blue}{1.999999999 \cdot 10^{-9} + \left(-0.7506356576441799 \cdot {x}^{3} + \left(-1.2734922589667106 \cdot {x}^{2} + 2.256772713883663 \cdot x\right)\right)}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]
   7. Step-by-step derivation

      1. +-commutative 99.6%

         \[\leadsto \frac{\color{blue}{\left(-0.7506356576441799 \cdot {x}^{3} + \left(-1.2734922589667106 \cdot {x}^{2} + 2.256772713883663 \cdot x\right)\right) + 1.999999999 \cdot 10^{-9}}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      2. associate-+r+ 99.6%

         \[\leadsto \frac{\color{blue}{\left(\left(-0.7506356576441799 \cdot {x}^{3} + -1.2734922589667106 \cdot {x}^{2}\right) + 2.256772713883663 \cdot x\right)} + 1.999999999 \cdot 10^{-9}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      3. associate-+l+ 99.6%

         \[\leadsto \frac{\color{blue}{\left(-0.7506356576441799 \cdot {x}^{3} + -1.2734922589667106 \cdot {x}^{2}\right) + \left(2.256772713883663 \cdot x + 1.999999999 \cdot 10^{-9}\right)}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      4. *-commutative 99.6%

         \[\leadsto \frac{\left(\color{blue}{{x}^{3} \cdot -0.7506356576441799} + -1.2734922589667106 \cdot {x}^{2}\right) + \left(2.256772713883663 \cdot x + 1.999999999 \cdot 10^{-9}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      5. fma-def 99.6%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, -1.2734922589667106 \cdot {x}^{2}\right)} + \left(2.256772713883663 \cdot x + 1.999999999 \cdot 10^{-9}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      6. *-commutative 99.6%

         \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, \color{blue}{{x}^{2} \cdot -1.2734922589667106}\right) + \left(2.256772713883663 \cdot x + 1.999999999 \cdot 10^{-9}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      7. unpow2 99.6%

         \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, \color{blue}{\left(x \cdot x\right)} \cdot -1.2734922589667106\right) + \left(2.256772713883663 \cdot x + 1.999999999 \cdot 10^{-9}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      8. associate-*l* 99.6%

         \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, \color{blue}{x \cdot \left(x \cdot -1.2734922589667106\right)}\right) + \left(2.256772713883663 \cdot x + 1.999999999 \cdot 10^{-9}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      9. *-commutative 99.6%

         \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, x \cdot \left(x \cdot -1.2734922589667106\right)\right) + \left(\color{blue}{x \cdot 2.256772713883663} + 1.999999999 \cdot 10^{-9}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      10. fma-def 99.6%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, x \cdot \left(x \cdot -1.2734922589667106\right)\right) + \color{blue}{\mathsf{fma}\left(x, 2.256772713883663, 1.999999999 \cdot 10^{-9}\right)}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   8. Simplified 99.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, x \cdot \left(x \cdot -1.2734922589667106\right)\right) + \mathsf{fma}\left(x, 2.256772713883663, 1.999999999 \cdot 10^{-9}\right)}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   if 4.9999999999999997e-12 < (fabs.f64 x)

   1. Initial program 99.1%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]
   4. Step-by-step derivation

      1. expm1-log1p-u 99.1%

         \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]

   5. Applied egg-rr 99.1%

      \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, x \cdot \left(x \cdot -1.2734922589667106\right)\right) + \mathsf{fma}\left(x, 2.256772713883663, 1.999999999 \cdot 10^{-9}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.254829592 + \mathsf{expm1}\left(\mathsf{log1p}\left(\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}\\ \end{array} \]

Alternative 2: 99.0% accurate, 0.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)\\ \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, x \cdot \left(x \cdot -1.2734922589667106\right)\right) + \mathsf{fma}\left(x, 2.256772713883663, 1.999999999 \cdot 10^{-9}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(1 + e^{x \cdot \left(-x\right)} \cdot \frac{\frac{\frac{\frac{1.453152027 + 1.061405429 \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911}}{t_0} - 1.421413741}{t_0} - -0.284496736}{t_0} - 0.254829592}{t_0}\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (fma 0.3275911 (fabs x) 1.0)))
   (if (<= (fabs x) 5e-12)
     (/
      (+
       (fma (pow x 3.0) -0.7506356576441799 (* x (* x -1.2734922589667106)))
       (fma x 2.256772713883663 1.999999999e-9))
      (+
       1.0
       (/
        (+
         0.254829592
         (/
          (+
           -0.284496736
           (/
            (+
             1.421413741
             (/
              (+ -1.453152027 (/ 1.061405429 (fma x 0.3275911 1.0)))
              (fma x 0.3275911 1.0)))
            (fma x 0.3275911 1.0)))
          (fma x 0.3275911 1.0)))
        (* (fma x 0.3275911 1.0) (exp (* x x))))))
     (exp
      (log
       (+
        1.0
        (*
         (exp (* x (- x)))
         (/
          (-
           (/
            (-
             (/
              (-
               (/
                (+
                 1.453152027
                 (* 1.061405429 (/ -1.0 (+ 1.0 (* (fabs x) 0.3275911)))))
                t_0)
               1.421413741)
              t_0)
             -0.284496736)
            t_0)
           0.254829592)
          t_0))))))))

C

double code(double x) {
	double t_0 = fma(0.3275911, fabs(x), 1.0);
	double tmp;
	if (fabs(x) <= 5e-12) {
		tmp = (fma(pow(x, 3.0), -0.7506356576441799, (x * (x * -1.2734922589667106))) + fma(x, 2.256772713883663, 1.999999999e-9)) / (1.0 + ((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / (fma(x, 0.3275911, 1.0) * exp((x * x)))));
	} else {
		tmp = exp(log((1.0 + (exp((x * -x)) * ((((((((1.453152027 + (1.061405429 * (-1.0 / (1.0 + (fabs(x) * 0.3275911))))) / t_0) - 1.421413741) / t_0) - -0.284496736) / t_0) - 0.254829592) / t_0)))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = fma(0.3275911, abs(x), 1.0)
	tmp = 0.0
	if (abs(x) <= 5e-12)
		tmp = Float64(Float64(fma((x ^ 3.0), -0.7506356576441799, Float64(x * Float64(x * -1.2734922589667106))) + fma(x, 2.256772713883663, 1.999999999e-9)) / Float64(1.0 + Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / Float64(fma(x, 0.3275911, 1.0) * exp(Float64(x * x))))));
	else
		tmp = exp(log(Float64(1.0 + Float64(exp(Float64(x * Float64(-x))) * Float64(Float64(Float64(Float64(Float64(Float64(Float64(Float64(1.453152027 + Float64(1.061405429 * Float64(-1.0 / Float64(1.0 + Float64(abs(x) * 0.3275911))))) / t_0) - 1.421413741) / t_0) - -0.284496736) / t_0) - 0.254829592) / t_0)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 5e-12], N[(N[(N[(N[Power[x, 3.0], $MachinePrecision] * -0.7506356576441799 + N[(x * N[(x * -1.2734922589667106), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 2.256772713883663 + 1.999999999e-9), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * 0.3275911 + 1.0), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[Exp[N[Log[N[(1.0 + N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] * N[(N[(N[(N[(N[(N[(N[(N[(1.453152027 + N[(1.061405429 * N[(-1.0 / N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision] - 1.421413741), $MachinePrecision] / t$95$0), $MachinePrecision] - -0.284496736), $MachinePrecision] / t$95$0), $MachinePrecision] - 0.254829592), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]], $MachinePrecision]], $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 4.9999999999999997e-12

   1. Initial program 57.7%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Applied egg-rr 57.7%

      \[\leadsto \color{blue}{\frac{1 - \left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}\right) \cdot \left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}}} \]

   5. Simplified 57.7%

      \[\leadsto \color{blue}{\frac{1 + \frac{\frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{e^{x \cdot x}} \cdot \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}} \]

   6. Taylor expanded in x around 0 99.6%

      \[\leadsto \frac{\color{blue}{1.999999999 \cdot 10^{-9} + \left(-0.7506356576441799 \cdot {x}^{3} + \left(-1.2734922589667106 \cdot {x}^{2} + 2.256772713883663 \cdot x\right)\right)}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]
   7. Step-by-step derivation

      1. +-commutative 99.6%

         \[\leadsto \frac{\color{blue}{\left(-0.7506356576441799 \cdot {x}^{3} + \left(-1.2734922589667106 \cdot {x}^{2} + 2.256772713883663 \cdot x\right)\right) + 1.999999999 \cdot 10^{-9}}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      2. associate-+r+ 99.6%

         \[\leadsto \frac{\color{blue}{\left(\left(-0.7506356576441799 \cdot {x}^{3} + -1.2734922589667106 \cdot {x}^{2}\right) + 2.256772713883663 \cdot x\right)} + 1.999999999 \cdot 10^{-9}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      3. associate-+l+ 99.6%

         \[\leadsto \frac{\color{blue}{\left(-0.7506356576441799 \cdot {x}^{3} + -1.2734922589667106 \cdot {x}^{2}\right) + \left(2.256772713883663 \cdot x + 1.999999999 \cdot 10^{-9}\right)}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      4. *-commutative 99.6%

         \[\leadsto \frac{\left(\color{blue}{{x}^{3} \cdot -0.7506356576441799} + -1.2734922589667106 \cdot {x}^{2}\right) + \left(2.256772713883663 \cdot x + 1.999999999 \cdot 10^{-9}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      5. fma-def 99.6%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, -1.2734922589667106 \cdot {x}^{2}\right)} + \left(2.256772713883663 \cdot x + 1.999999999 \cdot 10^{-9}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      6. *-commutative 99.6%

         \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, \color{blue}{{x}^{2} \cdot -1.2734922589667106}\right) + \left(2.256772713883663 \cdot x + 1.999999999 \cdot 10^{-9}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      7. unpow2 99.6%

         \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, \color{blue}{\left(x \cdot x\right)} \cdot -1.2734922589667106\right) + \left(2.256772713883663 \cdot x + 1.999999999 \cdot 10^{-9}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      8. associate-*l* 99.6%

         \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, \color{blue}{x \cdot \left(x \cdot -1.2734922589667106\right)}\right) + \left(2.256772713883663 \cdot x + 1.999999999 \cdot 10^{-9}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      9. *-commutative 99.6%

         \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, x \cdot \left(x \cdot -1.2734922589667106\right)\right) + \left(\color{blue}{x \cdot 2.256772713883663} + 1.999999999 \cdot 10^{-9}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      10. fma-def 99.6%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, x \cdot \left(x \cdot -1.2734922589667106\right)\right) + \color{blue}{\mathsf{fma}\left(x, 2.256772713883663, 1.999999999 \cdot 10^{-9}\right)}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   8. Simplified 99.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, x \cdot \left(x \cdot -1.2734922589667106\right)\right) + \mathsf{fma}\left(x, 2.256772713883663, 1.999999999 \cdot 10^{-9}\right)}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   if 4.9999999999999997e-12 < (fabs.f64 x)

   1. Initial program 99.1%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Applied egg-rr 99.1%

      \[\leadsto \color{blue}{e^{\log \left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}\right)}} \]

   4. Taylor expanded in x around inf 99.1%

      \[\leadsto e^{\log \left(1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{\color{blue}{1.061405429 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1} - 1.453152027}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, x \cdot \left(x \cdot -1.2734922589667106\right)\right) + \mathsf{fma}\left(x, 2.256772713883663, 1.999999999 \cdot 10^{-9}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;e^{\log \left(1 + e^{x \cdot \left(-x\right)} \cdot \frac{\frac{\frac{\frac{1.453152027 + 1.061405429 \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 1.421413741}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - -0.284496736}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} - 0.254829592}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}\\ \end{array} \]

Alternative 3: 99.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, x \cdot \left(x \cdot -1.2734922589667106\right)\right) + \mathsf{fma}\left(x, 2.256772713883663, 1.999999999 \cdot 10^{-9}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.254829592 + \frac{\left(1.061405429 \cdot \frac{1}{{t_0}^{3}} + 1.421413741 \cdot \frac{1}{t_0}\right) + \left(1.453152027 \cdot \frac{-1}{{t_0}^{2}} - 0.284496736\right)}{t_0}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x) 0.3275911))))
   (if (<= (fabs x) 5e-12)
     (/
      (+
       (fma (pow x 3.0) -0.7506356576441799 (* x (* x -1.2734922589667106)))
       (fma x 2.256772713883663 1.999999999e-9))
      (+
       1.0
       (/
        (+
         0.254829592
         (/
          (+
           -0.284496736
           (/
            (+
             1.421413741
             (/
              (+ -1.453152027 (/ 1.061405429 (fma x 0.3275911 1.0)))
              (fma x 0.3275911 1.0)))
            (fma x 0.3275911 1.0)))
          (fma x 0.3275911 1.0)))
        (* (fma x 0.3275911 1.0) (exp (* x x))))))
     (-
      1.0
      (/
       (+
        0.254829592
        (/
         (+
          (+ (* 1.061405429 (/ 1.0 (pow t_0 3.0))) (* 1.421413741 (/ 1.0 t_0)))
          (- (* 1.453152027 (/ -1.0 (pow t_0 2.0))) 0.284496736))
         t_0))
       (* (fma 0.3275911 (fabs x) 1.0) (pow (exp x) x)))))))

C

double code(double x) {
	double t_0 = 1.0 + (fabs(x) * 0.3275911);
	double tmp;
	if (fabs(x) <= 5e-12) {
		tmp = (fma(pow(x, 3.0), -0.7506356576441799, (x * (x * -1.2734922589667106))) + fma(x, 2.256772713883663, 1.999999999e-9)) / (1.0 + ((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / (fma(x, 0.3275911, 1.0) * exp((x * x)))));
	} else {
		tmp = 1.0 - ((0.254829592 + ((((1.061405429 * (1.0 / pow(t_0, 3.0))) + (1.421413741 * (1.0 / t_0))) + ((1.453152027 * (-1.0 / pow(t_0, 2.0))) - 0.284496736)) / t_0)) / (fma(0.3275911, fabs(x), 1.0) * pow(exp(x), x)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	tmp = 0.0
	if (abs(x) <= 5e-12)
		tmp = Float64(Float64(fma((x ^ 3.0), -0.7506356576441799, Float64(x * Float64(x * -1.2734922589667106))) + fma(x, 2.256772713883663, 1.999999999e-9)) / Float64(1.0 + Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / Float64(fma(x, 0.3275911, 1.0) * exp(Float64(x * x))))));
	else
		tmp = Float64(1.0 - Float64(Float64(0.254829592 + Float64(Float64(Float64(Float64(1.061405429 * Float64(1.0 / (t_0 ^ 3.0))) + Float64(1.421413741 * Float64(1.0 / t_0))) + Float64(Float64(1.453152027 * Float64(-1.0 / (t_0 ^ 2.0))) - 0.284496736)) / t_0)) / Float64(fma(0.3275911, abs(x), 1.0) * (exp(x) ^ x))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 5e-12], N[(N[(N[(N[Power[x, 3.0], $MachinePrecision] * -0.7506356576441799 + N[(x * N[(x * -1.2734922589667106), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 2.256772713883663 + 1.999999999e-9), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * 0.3275911 + 1.0), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(N[(0.254829592 + N[(N[(N[(N[(1.061405429 * N[(1.0 / N[Power[t$95$0, 3.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(1.421413741 * N[(1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(1.453152027 * N[(-1.0 / N[Power[t$95$0, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] - 0.284496736), $MachinePrecision]), $MachinePrecision] / t$95$0), $MachinePrecision]), $MachinePrecision] / N[(N[(0.3275911 * N[Abs[x], $MachinePrecision] + 1.0), $MachinePrecision] * N[Power[N[Exp[x], $MachinePrecision], x], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 4.9999999999999997e-12

   1. Initial program 57.7%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Applied egg-rr 57.7%

      \[\leadsto \color{blue}{\frac{1 - \left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}\right) \cdot \left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}}} \]

   5. Simplified 57.7%

      \[\leadsto \color{blue}{\frac{1 + \frac{\frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{e^{x \cdot x}} \cdot \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}} \]

   6. Taylor expanded in x around 0 99.6%

      \[\leadsto \frac{\color{blue}{1.999999999 \cdot 10^{-9} + \left(-0.7506356576441799 \cdot {x}^{3} + \left(-1.2734922589667106 \cdot {x}^{2} + 2.256772713883663 \cdot x\right)\right)}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]
   7. Step-by-step derivation

      1. +-commutative 99.6%

         \[\leadsto \frac{\color{blue}{\left(-0.7506356576441799 \cdot {x}^{3} + \left(-1.2734922589667106 \cdot {x}^{2} + 2.256772713883663 \cdot x\right)\right) + 1.999999999 \cdot 10^{-9}}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      2. associate-+r+ 99.6%

         \[\leadsto \frac{\color{blue}{\left(\left(-0.7506356576441799 \cdot {x}^{3} + -1.2734922589667106 \cdot {x}^{2}\right) + 2.256772713883663 \cdot x\right)} + 1.999999999 \cdot 10^{-9}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      3. associate-+l+ 99.6%

         \[\leadsto \frac{\color{blue}{\left(-0.7506356576441799 \cdot {x}^{3} + -1.2734922589667106 \cdot {x}^{2}\right) + \left(2.256772713883663 \cdot x + 1.999999999 \cdot 10^{-9}\right)}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      4. *-commutative 99.6%

         \[\leadsto \frac{\left(\color{blue}{{x}^{3} \cdot -0.7506356576441799} + -1.2734922589667106 \cdot {x}^{2}\right) + \left(2.256772713883663 \cdot x + 1.999999999 \cdot 10^{-9}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      5. fma-def 99.6%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, -1.2734922589667106 \cdot {x}^{2}\right)} + \left(2.256772713883663 \cdot x + 1.999999999 \cdot 10^{-9}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      6. *-commutative 99.6%

         \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, \color{blue}{{x}^{2} \cdot -1.2734922589667106}\right) + \left(2.256772713883663 \cdot x + 1.999999999 \cdot 10^{-9}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      7. unpow2 99.6%

         \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, \color{blue}{\left(x \cdot x\right)} \cdot -1.2734922589667106\right) + \left(2.256772713883663 \cdot x + 1.999999999 \cdot 10^{-9}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      8. associate-*l* 99.6%

         \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, \color{blue}{x \cdot \left(x \cdot -1.2734922589667106\right)}\right) + \left(2.256772713883663 \cdot x + 1.999999999 \cdot 10^{-9}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      9. *-commutative 99.6%

         \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, x \cdot \left(x \cdot -1.2734922589667106\right)\right) + \left(\color{blue}{x \cdot 2.256772713883663} + 1.999999999 \cdot 10^{-9}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      10. fma-def 99.6%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, x \cdot \left(x \cdot -1.2734922589667106\right)\right) + \color{blue}{\mathsf{fma}\left(x, 2.256772713883663, 1.999999999 \cdot 10^{-9}\right)}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   8. Simplified 99.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, x \cdot \left(x \cdot -1.2734922589667106\right)\right) + \mathsf{fma}\left(x, 2.256772713883663, 1.999999999 \cdot 10^{-9}\right)}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   if 4.9999999999999997e-12 < (fabs.f64 x)

   1. Initial program 99.1%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   4. Taylor expanded in x around inf 99.1%

      \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\frac{\left(1.061405429 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{3}} + 1.421413741 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right) - \left(0.284496736 + 1.453152027 \cdot \frac{1}{{\left(0.3275911 \cdot \left|x\right| + 1\right)}^{2}}\right)}{0.3275911 \cdot \left|x\right| + 1}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, x \cdot \left(x \cdot -1.2734922589667106\right)\right) + \mathsf{fma}\left(x, 2.256772713883663, 1.999999999 \cdot 10^{-9}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.254829592 + \frac{\left(1.061405429 \cdot \frac{1}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^{3}} + 1.421413741 \cdot \frac{1}{1 + \left|x\right| \cdot 0.3275911}\right) + \left(1.453152027 \cdot \frac{-1}{{\left(1 + \left|x\right| \cdot 0.3275911\right)}^{2}} - 0.284496736\right)}{1 + \left|x\right| \cdot 0.3275911}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}\\ \end{array} \]

Alternative 4: 99.0% accurate, 0.8× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, x \cdot \left(x \cdot -1.2734922589667106\right)\right) + \mathsf{fma}\left(x, 2.256772713883663, 1.999999999 \cdot 10^{-9}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(t_1 \cdot \left(t_1 \cdot \left(t_1 \cdot \left(t_1 \cdot \left(1.061405429 \cdot \frac{-1}{t_0} - -1.453152027\right) - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x) 0.3275911))) (t_1 (/ 1.0 t_0)))
   (if (<= (fabs x) 5e-12)
     (/
      (+
       (fma (pow x 3.0) -0.7506356576441799 (* x (* x -1.2734922589667106)))
       (fma x 2.256772713883663 1.999999999e-9))
      (+
       1.0
       (/
        (+
         0.254829592
         (/
          (+
           -0.284496736
           (/
            (+
             1.421413741
             (/
              (+ -1.453152027 (/ 1.061405429 (fma x 0.3275911 1.0)))
              (fma x 0.3275911 1.0)))
            (fma x 0.3275911 1.0)))
          (fma x 0.3275911 1.0)))
        (* (fma x 0.3275911 1.0) (exp (* x x))))))
     (+
      1.0
      (*
       (exp (* x (- x)))
       (*
        t_1
        (-
         (*
          t_1
          (-
           (*
            t_1
            (-
             (* t_1 (- (* 1.061405429 (/ -1.0 t_0)) -1.453152027))
             1.421413741))
           -0.284496736))
         0.254829592)))))))

C

double code(double x) {
	double t_0 = 1.0 + (fabs(x) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (fabs(x) <= 5e-12) {
		tmp = (fma(pow(x, 3.0), -0.7506356576441799, (x * (x * -1.2734922589667106))) + fma(x, 2.256772713883663, 1.999999999e-9)) / (1.0 + ((0.254829592 + ((-0.284496736 + ((1.421413741 + ((-1.453152027 + (1.061405429 / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / (fma(x, 0.3275911, 1.0) * exp((x * x)))));
	} else {
		tmp = 1.0 + (exp((x * -x)) * (t_1 * ((t_1 * ((t_1 * ((t_1 * ((1.061405429 * (-1.0 / t_0)) - -1.453152027)) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (abs(x) <= 5e-12)
		tmp = Float64(Float64(fma((x ^ 3.0), -0.7506356576441799, Float64(x * Float64(x * -1.2734922589667106))) + fma(x, 2.256772713883663, 1.999999999e-9)) / Float64(1.0 + Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / Float64(fma(x, 0.3275911, 1.0) * exp(Float64(x * x))))));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x * Float64(-x))) * Float64(t_1 * Float64(Float64(t_1 * Float64(Float64(t_1 * Float64(Float64(t_1 * Float64(Float64(1.061405429 * Float64(-1.0 / t_0)) - -1.453152027)) - 1.421413741)) - -0.284496736)) - 0.254829592))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 5e-12], N[(N[(N[(N[Power[x, 3.0], $MachinePrecision] * -0.7506356576441799 + N[(x * N[(x * -1.2734922589667106), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(x * 2.256772713883663 + 1.999999999e-9), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * 0.3275911 + 1.0), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[(N[(t$95$1 * N[(N[(t$95$1 * N[(N[(t$95$1 * N[(N[(1.061405429 * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] - -1.453152027), $MachinePrecision]), $MachinePrecision] - 1.421413741), $MachinePrecision]), $MachinePrecision] - -0.284496736), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 4.9999999999999997e-12

   1. Initial program 57.7%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Applied egg-rr 57.7%

      \[\leadsto \color{blue}{\frac{1 - \left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}\right) \cdot \left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}}} \]

   5. Simplified 57.7%

      \[\leadsto \color{blue}{\frac{1 + \frac{\frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{e^{x \cdot x}} \cdot \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}} \]

   6. Taylor expanded in x around 0 99.6%

      \[\leadsto \frac{\color{blue}{1.999999999 \cdot 10^{-9} + \left(-0.7506356576441799 \cdot {x}^{3} + \left(-1.2734922589667106 \cdot {x}^{2} + 2.256772713883663 \cdot x\right)\right)}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]
   7. Step-by-step derivation

      1. +-commutative 99.6%

         \[\leadsto \frac{\color{blue}{\left(-0.7506356576441799 \cdot {x}^{3} + \left(-1.2734922589667106 \cdot {x}^{2} + 2.256772713883663 \cdot x\right)\right) + 1.999999999 \cdot 10^{-9}}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      2. associate-+r+ 99.6%

         \[\leadsto \frac{\color{blue}{\left(\left(-0.7506356576441799 \cdot {x}^{3} + -1.2734922589667106 \cdot {x}^{2}\right) + 2.256772713883663 \cdot x\right)} + 1.999999999 \cdot 10^{-9}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      3. associate-+l+ 99.6%

         \[\leadsto \frac{\color{blue}{\left(-0.7506356576441799 \cdot {x}^{3} + -1.2734922589667106 \cdot {x}^{2}\right) + \left(2.256772713883663 \cdot x + 1.999999999 \cdot 10^{-9}\right)}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      4. *-commutative 99.6%

         \[\leadsto \frac{\left(\color{blue}{{x}^{3} \cdot -0.7506356576441799} + -1.2734922589667106 \cdot {x}^{2}\right) + \left(2.256772713883663 \cdot x + 1.999999999 \cdot 10^{-9}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      5. fma-def 99.6%

         \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, -1.2734922589667106 \cdot {x}^{2}\right)} + \left(2.256772713883663 \cdot x + 1.999999999 \cdot 10^{-9}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      6. *-commutative 99.6%

         \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, \color{blue}{{x}^{2} \cdot -1.2734922589667106}\right) + \left(2.256772713883663 \cdot x + 1.999999999 \cdot 10^{-9}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      7. unpow2 99.6%

         \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, \color{blue}{\left(x \cdot x\right)} \cdot -1.2734922589667106\right) + \left(2.256772713883663 \cdot x + 1.999999999 \cdot 10^{-9}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      8. associate-*l* 99.6%

         \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, \color{blue}{x \cdot \left(x \cdot -1.2734922589667106\right)}\right) + \left(2.256772713883663 \cdot x + 1.999999999 \cdot 10^{-9}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      9. *-commutative 99.6%

         \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, x \cdot \left(x \cdot -1.2734922589667106\right)\right) + \left(\color{blue}{x \cdot 2.256772713883663} + 1.999999999 \cdot 10^{-9}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

      10. fma-def 99.6%

          \[\leadsto \frac{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, x \cdot \left(x \cdot -1.2734922589667106\right)\right) + \color{blue}{\mathsf{fma}\left(x, 2.256772713883663, 1.999999999 \cdot 10^{-9}\right)}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   8. Simplified 99.6%

      \[\leadsto \frac{\color{blue}{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, x \cdot \left(x \cdot -1.2734922589667106\right)\right) + \mathsf{fma}\left(x, 2.256772713883663, 1.999999999 \cdot 10^{-9}\right)}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   if 4.9999999999999997e-12 < (fabs.f64 x)

   1. Initial program 99.1%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{\mathsf{fma}\left({x}^{3}, -0.7506356576441799, x \cdot \left(x \cdot -1.2734922589667106\right)\right) + \mathsf{fma}\left(x, 2.256772713883663, 1.999999999 \cdot 10^{-9}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.061405429 \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911} - -1.453152027\right) - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \]

Alternative 5: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{1.999999999 \cdot 10^{-9} + \left({x}^{3} \cdot -0.7506356576441799 + \left(-1.2734922589667106 \cdot {x}^{2} + x \cdot 2.256772713883663\right)\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.029667143 + x \cdot -0.2193742730720041}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(t_1 \cdot \left(t_1 \cdot \left(t_1 \cdot \left(t_1 \cdot \left(1.061405429 \cdot \frac{-1}{t_0} - -1.453152027\right) - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x) 0.3275911))) (t_1 (/ 1.0 t_0)))
   (if (<= (fabs x) 5e-12)
     (/
      (+
       1.999999999e-9
       (+
        (* (pow x 3.0) -0.7506356576441799)
        (+ (* -1.2734922589667106 (pow x 2.0)) (* x 2.256772713883663))))
      (+
       1.0
       (/
        (+
         0.254829592
         (/
          (+
           -0.284496736
           (/ (+ 1.029667143 (* x -0.2193742730720041)) (fma x 0.3275911 1.0)))
          (fma x 0.3275911 1.0)))
        (* (fma x 0.3275911 1.0) (exp (* x x))))))
     (+
      1.0
      (*
       (exp (* x (- x)))
       (*
        t_1
        (-
         (*
          t_1
          (-
           (*
            t_1
            (-
             (* t_1 (- (* 1.061405429 (/ -1.0 t_0)) -1.453152027))
             1.421413741))
           -0.284496736))
         0.254829592)))))))

C

double code(double x) {
	double t_0 = 1.0 + (fabs(x) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (fabs(x) <= 5e-12) {
		tmp = (1.999999999e-9 + ((pow(x, 3.0) * -0.7506356576441799) + ((-1.2734922589667106 * pow(x, 2.0)) + (x * 2.256772713883663)))) / (1.0 + ((0.254829592 + ((-0.284496736 + ((1.029667143 + (x * -0.2193742730720041)) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / (fma(x, 0.3275911, 1.0) * exp((x * x)))));
	} else {
		tmp = 1.0 + (exp((x * -x)) * (t_1 * ((t_1 * ((t_1 * ((t_1 * ((1.061405429 * (-1.0 / t_0)) - -1.453152027)) - 1.421413741)) - -0.284496736)) - 0.254829592)));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (abs(x) <= 5e-12)
		tmp = Float64(Float64(1.999999999e-9 + Float64(Float64((x ^ 3.0) * -0.7506356576441799) + Float64(Float64(-1.2734922589667106 * (x ^ 2.0)) + Float64(x * 2.256772713883663)))) / Float64(1.0 + Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.029667143 + Float64(x * -0.2193742730720041)) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / Float64(fma(x, 0.3275911, 1.0) * exp(Float64(x * x))))));
	else
		tmp = Float64(1.0 + Float64(exp(Float64(x * Float64(-x))) * Float64(t_1 * Float64(Float64(t_1 * Float64(Float64(t_1 * Float64(Float64(t_1 * Float64(Float64(1.061405429 * Float64(-1.0 / t_0)) - -1.453152027)) - 1.421413741)) - -0.284496736)) - 0.254829592))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 5e-12], N[(N[(1.999999999e-9 + N[(N[(N[Power[x, 3.0], $MachinePrecision] * -0.7506356576441799), $MachinePrecision] + N[(N[(-1.2734922589667106 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x * 2.256772713883663), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.029667143 + N[(x * -0.2193742730720041), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * 0.3275911 + 1.0), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] * N[(t$95$1 * N[(N[(t$95$1 * N[(N[(t$95$1 * N[(N[(t$95$1 * N[(N[(1.061405429 * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision] - -1.453152027), $MachinePrecision]), $MachinePrecision] - 1.421413741), $MachinePrecision]), $MachinePrecision] - -0.284496736), $MachinePrecision]), $MachinePrecision] - 0.254829592), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 4.9999999999999997e-12

   1. Initial program 57.7%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Applied egg-rr 57.7%

      \[\leadsto \color{blue}{\frac{1 - \left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}\right) \cdot \left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}}} \]

   5. Simplified 57.7%

      \[\leadsto \color{blue}{\frac{1 + \frac{\frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{e^{x \cdot x}} \cdot \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}} \]

   6. Taylor expanded in x around 0 99.6%

      \[\leadsto \frac{\color{blue}{1.999999999 \cdot 10^{-9} + \left(-0.7506356576441799 \cdot {x}^{3} + \left(-1.2734922589667106 \cdot {x}^{2} + 2.256772713883663 \cdot x\right)\right)}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   7. Taylor expanded in x around 0 99.6%

      \[\leadsto \frac{1.999999999 \cdot 10^{-9} + \left(-0.7506356576441799 \cdot {x}^{3} + \left(-1.2734922589667106 \cdot {x}^{2} + 2.256772713883663 \cdot x\right)\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{\color{blue}{1.029667143 + -0.2193742730720041 \cdot x}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]
   8. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \frac{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.029667143 + \color{blue}{x \cdot -0.2193742730720041}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   9. Simplified 99.6%

      \[\leadsto \frac{1.999999999 \cdot 10^{-9} + \left(-0.7506356576441799 \cdot {x}^{3} + \left(-1.2734922589667106 \cdot {x}^{2} + 2.256772713883663 \cdot x\right)\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{\color{blue}{1.029667143 + x \cdot -0.2193742730720041}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   if 4.9999999999999997e-12 < (fabs.f64 x)

   1. Initial program 99.1%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{1.999999999 \cdot 10^{-9} + \left({x}^{3} \cdot -0.7506356576441799 + \left(-1.2734922589667106 \cdot {x}^{2} + x \cdot 2.256772713883663\right)\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.029667143 + x \cdot -0.2193742730720041}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + e^{x \cdot \left(-x\right)} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(\frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.061405429 \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911} - -1.453152027\right) - 1.421413741\right) - -0.284496736\right) - 0.254829592\right)\right)\\ \end{array} \]

Alternative 6: 99.0% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{1.999999999 \cdot 10^{-9} + \left({x}^{3} \cdot -0.7506356576441799 + \left(-1.2734922589667106 \cdot {x}^{2} + x \cdot 2.256772713883663\right)\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.029667143 + x \cdot -0.2193742730720041}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(e^{x \cdot \left(-x\right)} \cdot \left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_1 \cdot \left(1.421413741 + t_1 \cdot \left(-1.453152027 + \frac{1.061405429}{t_0}\right)\right)\right)\right)\right) \cdot \frac{-1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x) 0.3275911))) (t_1 (/ 1.0 t_0)))
   (if (<= (fabs x) 5e-12)
     (/
      (+
       1.999999999e-9
       (+
        (* (pow x 3.0) -0.7506356576441799)
        (+ (* -1.2734922589667106 (pow x 2.0)) (* x 2.256772713883663))))
      (+
       1.0
       (/
        (+
         0.254829592
         (/
          (+
           -0.284496736
           (/ (+ 1.029667143 (* x -0.2193742730720041)) (fma x 0.3275911 1.0)))
          (fma x 0.3275911 1.0)))
        (* (fma x 0.3275911 1.0) (exp (* x x))))))
     (+
      1.0
      (*
       (*
        (exp (* x (- x)))
        (+
         0.254829592
         (*
          t_1
          (+
           -0.284496736
           (*
            t_1
            (+ 1.421413741 (* t_1 (+ -1.453152027 (/ 1.061405429 t_0)))))))))
       (/ -1.0 t_0))))))

C

double code(double x) {
	double t_0 = 1.0 + (fabs(x) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (fabs(x) <= 5e-12) {
		tmp = (1.999999999e-9 + ((pow(x, 3.0) * -0.7506356576441799) + ((-1.2734922589667106 * pow(x, 2.0)) + (x * 2.256772713883663)))) / (1.0 + ((0.254829592 + ((-0.284496736 + ((1.029667143 + (x * -0.2193742730720041)) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / (fma(x, 0.3275911, 1.0) * exp((x * x)))));
	} else {
		tmp = 1.0 + ((exp((x * -x)) * (0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + (t_1 * (-1.453152027 + (1.061405429 / t_0))))))))) * (-1.0 / t_0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (abs(x) <= 5e-12)
		tmp = Float64(Float64(1.999999999e-9 + Float64(Float64((x ^ 3.0) * -0.7506356576441799) + Float64(Float64(-1.2734922589667106 * (x ^ 2.0)) + Float64(x * 2.256772713883663)))) / Float64(1.0 + Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.029667143 + Float64(x * -0.2193742730720041)) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / Float64(fma(x, 0.3275911, 1.0) * exp(Float64(x * x))))));
	else
		tmp = Float64(1.0 + Float64(Float64(exp(Float64(x * Float64(-x))) * Float64(0.254829592 + Float64(t_1 * Float64(-0.284496736 + Float64(t_1 * Float64(1.421413741 + Float64(t_1 * Float64(-1.453152027 + Float64(1.061405429 / t_0))))))))) * Float64(-1.0 / t_0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 5e-12], N[(N[(1.999999999e-9 + N[(N[(N[Power[x, 3.0], $MachinePrecision] * -0.7506356576441799), $MachinePrecision] + N[(N[(-1.2734922589667106 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x * 2.256772713883663), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.029667143 + N[(x * -0.2193742730720041), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * 0.3275911 + 1.0), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] * N[(0.254829592 + N[(t$95$1 * N[(-0.284496736 + N[(t$95$1 * N[(1.421413741 + N[(t$95$1 * N[(-1.453152027 + N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 4.9999999999999997e-12

   1. Initial program 57.7%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Applied egg-rr 57.7%

      \[\leadsto \color{blue}{\frac{1 - \left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}\right) \cdot \left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}}} \]

   5. Simplified 57.7%

      \[\leadsto \color{blue}{\frac{1 + \frac{\frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{e^{x \cdot x}} \cdot \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}} \]

   6. Taylor expanded in x around 0 99.6%

      \[\leadsto \frac{\color{blue}{1.999999999 \cdot 10^{-9} + \left(-0.7506356576441799 \cdot {x}^{3} + \left(-1.2734922589667106 \cdot {x}^{2} + 2.256772713883663 \cdot x\right)\right)}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   7. Taylor expanded in x around 0 99.6%

      \[\leadsto \frac{1.999999999 \cdot 10^{-9} + \left(-0.7506356576441799 \cdot {x}^{3} + \left(-1.2734922589667106 \cdot {x}^{2} + 2.256772713883663 \cdot x\right)\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{\color{blue}{1.029667143 + -0.2193742730720041 \cdot x}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]
   8. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \frac{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.029667143 + \color{blue}{x \cdot -0.2193742730720041}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   9. Simplified 99.6%

      \[\leadsto \frac{1.999999999 \cdot 10^{-9} + \left(-0.7506356576441799 \cdot {x}^{3} + \left(-1.2734922589667106 \cdot {x}^{2} + 2.256772713883663 \cdot x\right)\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{\color{blue}{1.029667143 + x \cdot -0.2193742730720041}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   if 4.9999999999999997e-12 < (fabs.f64 x)

   1. Initial program 99.1%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
   2. Step-by-step derivation

      1. associate-*l* 99.1%

         \[\leadsto 1 - \color{blue}{\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]
3. Recombined 2 regimes into one program.

4. Final simplification 99.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{1.999999999 \cdot 10^{-9} + \left({x}^{3} \cdot -0.7506356576441799 + \left(-1.2734922589667106 \cdot {x}^{2} + x \cdot 2.256772713883663\right)\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.029667143 + x \cdot -0.2193742730720041}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(e^{x \cdot \left(-x\right)} \cdot \left(0.254829592 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}\right)\right)\right)\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911}\\ \end{array} \]

Alternative 7: 98.5% accurate, 1.1× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{1.999999999 \cdot 10^{-9} + \left({x}^{3} \cdot -0.7506356576441799 + \left(-1.2734922589667106 \cdot {x}^{2} + x \cdot 2.256772713883663\right)\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.029667143 + x \cdot -0.2193742730720041}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(e^{x \cdot \left(-x\right)} \cdot \left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_1 \cdot \left(1.421413741 + \left(-1.453152027 + \frac{1.061405429}{t_0}\right) \cdot \frac{1}{1 + x \cdot 0.3275911}\right)\right)\right)\right) \cdot \frac{-1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x) 0.3275911))) (t_1 (/ 1.0 t_0)))
   (if (<= (fabs x) 5e-12)
     (/
      (+
       1.999999999e-9
       (+
        (* (pow x 3.0) -0.7506356576441799)
        (+ (* -1.2734922589667106 (pow x 2.0)) (* x 2.256772713883663))))
      (+
       1.0
       (/
        (+
         0.254829592
         (/
          (+
           -0.284496736
           (/ (+ 1.029667143 (* x -0.2193742730720041)) (fma x 0.3275911 1.0)))
          (fma x 0.3275911 1.0)))
        (* (fma x 0.3275911 1.0) (exp (* x x))))))
     (+
      1.0
      (*
       (*
        (exp (* x (- x)))
        (+
         0.254829592
         (*
          t_1
          (+
           -0.284496736
           (*
            t_1
            (+
             1.421413741
             (*
              (+ -1.453152027 (/ 1.061405429 t_0))
              (/ 1.0 (+ 1.0 (* x 0.3275911))))))))))
       (/ -1.0 t_0))))))

C

double code(double x) {
	double t_0 = 1.0 + (fabs(x) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (fabs(x) <= 5e-12) {
		tmp = (1.999999999e-9 + ((pow(x, 3.0) * -0.7506356576441799) + ((-1.2734922589667106 * pow(x, 2.0)) + (x * 2.256772713883663)))) / (1.0 + ((0.254829592 + ((-0.284496736 + ((1.029667143 + (x * -0.2193742730720041)) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / (fma(x, 0.3275911, 1.0) * exp((x * x)))));
	} else {
		tmp = 1.0 + ((exp((x * -x)) * (0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + ((-1.453152027 + (1.061405429 / t_0)) * (1.0 / (1.0 + (x * 0.3275911)))))))))) * (-1.0 / t_0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (abs(x) <= 5e-12)
		tmp = Float64(Float64(1.999999999e-9 + Float64(Float64((x ^ 3.0) * -0.7506356576441799) + Float64(Float64(-1.2734922589667106 * (x ^ 2.0)) + Float64(x * 2.256772713883663)))) / Float64(1.0 + Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.029667143 + Float64(x * -0.2193742730720041)) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / Float64(fma(x, 0.3275911, 1.0) * exp(Float64(x * x))))));
	else
		tmp = Float64(1.0 + Float64(Float64(exp(Float64(x * Float64(-x))) * Float64(0.254829592 + Float64(t_1 * Float64(-0.284496736 + Float64(t_1 * Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / t_0)) * Float64(1.0 / Float64(1.0 + Float64(x * 0.3275911)))))))))) * Float64(-1.0 / t_0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 5e-12], N[(N[(1.999999999e-9 + N[(N[(N[Power[x, 3.0], $MachinePrecision] * -0.7506356576441799), $MachinePrecision] + N[(N[(-1.2734922589667106 * N[Power[x, 2.0], $MachinePrecision]), $MachinePrecision] + N[(x * 2.256772713883663), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.029667143 + N[(x * -0.2193742730720041), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * 0.3275911 + 1.0), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] * N[(0.254829592 + N[(t$95$1 * N[(-0.284496736 + N[(t$95$1 * N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 4.9999999999999997e-12

   1. Initial program 57.7%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Applied egg-rr 57.7%

      \[\leadsto \color{blue}{\frac{1 - \left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}\right) \cdot \left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}}} \]

   5. Simplified 57.7%

      \[\leadsto \color{blue}{\frac{1 + \frac{\frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{e^{x \cdot x}} \cdot \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}} \]

   6. Taylor expanded in x around 0 99.6%

      \[\leadsto \frac{\color{blue}{1.999999999 \cdot 10^{-9} + \left(-0.7506356576441799 \cdot {x}^{3} + \left(-1.2734922589667106 \cdot {x}^{2} + 2.256772713883663 \cdot x\right)\right)}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   7. Taylor expanded in x around 0 99.6%

      \[\leadsto \frac{1.999999999 \cdot 10^{-9} + \left(-0.7506356576441799 \cdot {x}^{3} + \left(-1.2734922589667106 \cdot {x}^{2} + 2.256772713883663 \cdot x\right)\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{\color{blue}{1.029667143 + -0.2193742730720041 \cdot x}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]
   8. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \frac{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.029667143 + \color{blue}{x \cdot -0.2193742730720041}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   9. Simplified 99.6%

      \[\leadsto \frac{1.999999999 \cdot 10^{-9} + \left(-0.7506356576441799 \cdot {x}^{3} + \left(-1.2734922589667106 \cdot {x}^{2} + 2.256772713883663 \cdot x\right)\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{\color{blue}{1.029667143 + x \cdot -0.2193742730720041}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   if 4.9999999999999997e-12 < (fabs.f64 x)

   1. Initial program 99.1%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
   2. Step-by-step derivation

      1. associate-*l* 99.1%

         \[\leadsto 1 - \color{blue}{\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]
   4. Step-by-step derivation

      1. pow1 99.1%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   5. Applied egg-rr 99.1%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   6. Step-by-step derivation

      1. unpow1 99.1%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      2. *-commutative 99.1%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left|x\right| \cdot 0.3275911}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      3. unpow1 99.1%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left|\color{blue}{{x}^{1}}\right| \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      4. sqr-pow 54.5%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      5. fabs-sqr 54.5%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)} \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      6. sqr-pow 97.4%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{x}^{1}} \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      7. unpow1 97.4%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{x} \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   7. Simplified 97.4%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{x \cdot 0.3275911}} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{1.999999999 \cdot 10^{-9} + \left({x}^{3} \cdot -0.7506356576441799 + \left(-1.2734922589667106 \cdot {x}^{2} + x \cdot 2.256772713883663\right)\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.029667143 + x \cdot -0.2193742730720041}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(e^{x \cdot \left(-x\right)} \cdot \left(0.254829592 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \left(-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}\right) \cdot \frac{1}{1 + x \cdot 0.3275911}\right)\right)\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911}\\ \end{array} \]

Alternative 8: 98.5% accurate, 1.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.029667143 + x \cdot -0.2193742730720041}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(e^{x \cdot \left(-x\right)} \cdot \left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_1 \cdot \left(1.421413741 + \left(-1.453152027 + \frac{1.061405429}{t_0}\right) \cdot \frac{1}{1 + x \cdot 0.3275911}\right)\right)\right)\right) \cdot \frac{-1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x) 0.3275911))) (t_1 (/ 1.0 t_0)))
   (if (<= (fabs x) 5e-12)
     (/
      (+ 1.999999999e-9 (* x 2.256772713883663))
      (+
       1.0
       (/
        (+
         0.254829592
         (/
          (+
           -0.284496736
           (/ (+ 1.029667143 (* x -0.2193742730720041)) (fma x 0.3275911 1.0)))
          (fma x 0.3275911 1.0)))
        (* (fma x 0.3275911 1.0) (exp (* x x))))))
     (+
      1.0
      (*
       (*
        (exp (* x (- x)))
        (+
         0.254829592
         (*
          t_1
          (+
           -0.284496736
           (*
            t_1
            (+
             1.421413741
             (*
              (+ -1.453152027 (/ 1.061405429 t_0))
              (/ 1.0 (+ 1.0 (* x 0.3275911))))))))))
       (/ -1.0 t_0))))))

C

double code(double x) {
	double t_0 = 1.0 + (fabs(x) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double tmp;
	if (fabs(x) <= 5e-12) {
		tmp = (1.999999999e-9 + (x * 2.256772713883663)) / (1.0 + ((0.254829592 + ((-0.284496736 + ((1.029667143 + (x * -0.2193742730720041)) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / (fma(x, 0.3275911, 1.0) * exp((x * x)))));
	} else {
		tmp = 1.0 + ((exp((x * -x)) * (0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + ((-1.453152027 + (1.061405429 / t_0)) * (1.0 / (1.0 + (x * 0.3275911)))))))))) * (-1.0 / t_0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	tmp = 0.0
	if (abs(x) <= 5e-12)
		tmp = Float64(Float64(1.999999999e-9 + Float64(x * 2.256772713883663)) / Float64(1.0 + Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.029667143 + Float64(x * -0.2193742730720041)) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / Float64(fma(x, 0.3275911, 1.0) * exp(Float64(x * x))))));
	else
		tmp = Float64(1.0 + Float64(Float64(exp(Float64(x * Float64(-x))) * Float64(0.254829592 + Float64(t_1 * Float64(-0.284496736 + Float64(t_1 * Float64(1.421413741 + Float64(Float64(-1.453152027 + Float64(1.061405429 / t_0)) * Float64(1.0 / Float64(1.0 + Float64(x * 0.3275911)))))))))) * Float64(-1.0 / t_0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 5e-12], N[(N[(1.999999999e-9 + N[(x * 2.256772713883663), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.029667143 + N[(x * -0.2193742730720041), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * 0.3275911 + 1.0), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] * N[(0.254829592 + N[(t$95$1 * N[(-0.284496736 + N[(t$95$1 * N[(1.421413741 + N[(N[(-1.453152027 + N[(1.061405429 / t$95$0), $MachinePrecision]), $MachinePrecision] * N[(1.0 / N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 4.9999999999999997e-12

   1. Initial program 57.7%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Applied egg-rr 57.7%

      \[\leadsto \color{blue}{\frac{1 - \left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}\right) \cdot \left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}}} \]

   5. Simplified 57.7%

      \[\leadsto \color{blue}{\frac{1 + \frac{\frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{e^{x \cdot x}} \cdot \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}} \]

   6. Taylor expanded in x around 0 99.6%

      \[\leadsto \frac{\color{blue}{1.999999999 \cdot 10^{-9} + 2.256772713883663 \cdot x}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]
   7. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \frac{1.999999999 \cdot 10^{-9} + \color{blue}{x \cdot 2.256772713883663}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   8. Simplified 99.6%

      \[\leadsto \frac{\color{blue}{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   9. Taylor expanded in x around 0 99.6%

      \[\leadsto \frac{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{\color{blue}{1.029667143 + -0.2193742730720041 \cdot x}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]
   10. Step-by-step derivation

       1. *-commutative 99.6%

          \[\leadsto \frac{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.029667143 + \color{blue}{x \cdot -0.2193742730720041}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   11. Simplified 99.6%

       \[\leadsto \frac{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{\color{blue}{1.029667143 + x \cdot -0.2193742730720041}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   if 4.9999999999999997e-12 < (fabs.f64 x)

   1. Initial program 99.1%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
   2. Step-by-step derivation

      1. associate-*l* 99.1%

         \[\leadsto 1 - \color{blue}{\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]
   4. Step-by-step derivation

      1. pow1 99.1%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   5. Applied egg-rr 99.1%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   6. Step-by-step derivation

      1. unpow1 99.1%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      2. *-commutative 99.1%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left|x\right| \cdot 0.3275911}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      3. unpow1 99.1%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left|\color{blue}{{x}^{1}}\right| \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      4. sqr-pow 54.5%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      5. fabs-sqr 54.5%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)} \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      6. sqr-pow 97.4%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{x}^{1}} \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      7. unpow1 97.4%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{x} \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   7. Simplified 97.4%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{x \cdot 0.3275911}} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.029667143 + x \cdot -0.2193742730720041}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(e^{x \cdot \left(-x\right)} \cdot \left(0.254829592 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \left(-1.453152027 + \frac{1.061405429}{1 + \left|x\right| \cdot 0.3275911}\right) \cdot \frac{1}{1 + x \cdot 0.3275911}\right)\right)\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911}\\ \end{array} \]

Alternative 9: 98.4% accurate, 1.5× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := 1 + \left|x\right| \cdot 0.3275911\\ t_1 := \frac{1}{t_0}\\ t_2 := 1 + x \cdot 0.3275911\\ \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.029667143 + x \cdot -0.2193742730720041}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(e^{x \cdot \left(-x\right)} \cdot \left(0.254829592 + t_1 \cdot \left(-0.284496736 + t_1 \cdot \left(1.421413741 + \frac{1}{t_2} \cdot \left(-1.453152027 + \frac{1.061405429}{t_2}\right)\right)\right)\right)\right) \cdot \frac{-1}{t_0}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (+ 1.0 (* (fabs x) 0.3275911)))
        (t_1 (/ 1.0 t_0))
        (t_2 (+ 1.0 (* x 0.3275911))))
   (if (<= (fabs x) 5e-12)
     (/
      (+ 1.999999999e-9 (* x 2.256772713883663))
      (+
       1.0
       (/
        (+
         0.254829592
         (/
          (+
           -0.284496736
           (/ (+ 1.029667143 (* x -0.2193742730720041)) (fma x 0.3275911 1.0)))
          (fma x 0.3275911 1.0)))
        (* (fma x 0.3275911 1.0) (exp (* x x))))))
     (+
      1.0
      (*
       (*
        (exp (* x (- x)))
        (+
         0.254829592
         (*
          t_1
          (+
           -0.284496736
           (*
            t_1
            (+
             1.421413741
             (* (/ 1.0 t_2) (+ -1.453152027 (/ 1.061405429 t_2)))))))))
       (/ -1.0 t_0))))))

C

double code(double x) {
	double t_0 = 1.0 + (fabs(x) * 0.3275911);
	double t_1 = 1.0 / t_0;
	double t_2 = 1.0 + (x * 0.3275911);
	double tmp;
	if (fabs(x) <= 5e-12) {
		tmp = (1.999999999e-9 + (x * 2.256772713883663)) / (1.0 + ((0.254829592 + ((-0.284496736 + ((1.029667143 + (x * -0.2193742730720041)) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / (fma(x, 0.3275911, 1.0) * exp((x * x)))));
	} else {
		tmp = 1.0 + ((exp((x * -x)) * (0.254829592 + (t_1 * (-0.284496736 + (t_1 * (1.421413741 + ((1.0 / t_2) * (-1.453152027 + (1.061405429 / t_2))))))))) * (-1.0 / t_0));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 + Float64(abs(x) * 0.3275911))
	t_1 = Float64(1.0 / t_0)
	t_2 = Float64(1.0 + Float64(x * 0.3275911))
	tmp = 0.0
	if (abs(x) <= 5e-12)
		tmp = Float64(Float64(1.999999999e-9 + Float64(x * 2.256772713883663)) / Float64(1.0 + Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.029667143 + Float64(x * -0.2193742730720041)) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / Float64(fma(x, 0.3275911, 1.0) * exp(Float64(x * x))))));
	else
		tmp = Float64(1.0 + Float64(Float64(exp(Float64(x * Float64(-x))) * Float64(0.254829592 + Float64(t_1 * Float64(-0.284496736 + Float64(t_1 * Float64(1.421413741 + Float64(Float64(1.0 / t_2) * Float64(-1.453152027 + Float64(1.061405429 / t_2))))))))) * Float64(-1.0 / t_0)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 / t$95$0), $MachinePrecision]}, Block[{t$95$2 = N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 5e-12], N[(N[(1.999999999e-9 + N[(x * 2.256772713883663), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.029667143 + N[(x * -0.2193742730720041), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * 0.3275911 + 1.0), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] * N[(0.254829592 + N[(t$95$1 * N[(-0.284496736 + N[(t$95$1 * N[(1.421413741 + N[(N[(1.0 / t$95$2), $MachinePrecision] * N[(-1.453152027 + N[(1.061405429 / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 4.9999999999999997e-12

   1. Initial program 57.7%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Applied egg-rr 57.7%

      \[\leadsto \color{blue}{\frac{1 - \left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}\right) \cdot \left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}}} \]

   5. Simplified 57.7%

      \[\leadsto \color{blue}{\frac{1 + \frac{\frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{e^{x \cdot x}} \cdot \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}} \]

   6. Taylor expanded in x around 0 99.6%

      \[\leadsto \frac{\color{blue}{1.999999999 \cdot 10^{-9} + 2.256772713883663 \cdot x}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]
   7. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \frac{1.999999999 \cdot 10^{-9} + \color{blue}{x \cdot 2.256772713883663}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   8. Simplified 99.6%

      \[\leadsto \frac{\color{blue}{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   9. Taylor expanded in x around 0 99.6%

      \[\leadsto \frac{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{\color{blue}{1.029667143 + -0.2193742730720041 \cdot x}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]
   10. Step-by-step derivation

       1. *-commutative 99.6%

          \[\leadsto \frac{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.029667143 + \color{blue}{x \cdot -0.2193742730720041}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   11. Simplified 99.6%

       \[\leadsto \frac{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{\color{blue}{1.029667143 + x \cdot -0.2193742730720041}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   if 4.9999999999999997e-12 < (fabs.f64 x)

   1. Initial program 99.1%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
   2. Step-by-step derivation

      1. associate-*l* 99.1%

         \[\leadsto 1 - \color{blue}{\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]
   4. Step-by-step derivation

      1. pow1 99.1%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   5. Applied egg-rr 99.1%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   6. Step-by-step derivation

      1. unpow1 99.1%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      2. *-commutative 99.1%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left|x\right| \cdot 0.3275911}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      3. unpow1 99.1%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left|\color{blue}{{x}^{1}}\right| \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      4. sqr-pow 54.5%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      5. fabs-sqr 54.5%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)} \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      6. sqr-pow 97.4%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{x}^{1}} \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      7. unpow1 97.4%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{x} \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   7. Simplified 97.4%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{x \cdot 0.3275911}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   8. Step-by-step derivation

      1. pow1 99.1%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   9. Applied egg-rr 97.4%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   10. Step-by-step derivation

       1. unpow1 99.1%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       2. *-commutative 99.1%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left|x\right| \cdot 0.3275911}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       3. unpow1 99.1%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left|\color{blue}{{x}^{1}}\right| \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. sqr-pow 54.5%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       5. fabs-sqr 54.5%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)} \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       6. sqr-pow 97.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{x}^{1}} \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       7. unpow1 97.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{x} \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   11. Simplified 97.4%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{x \cdot 0.3275911}} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.029667143 + x \cdot -0.2193742730720041}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(e^{x \cdot \left(-x\right)} \cdot \left(0.254829592 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right)\right)\right)\right) \cdot \frac{-1}{1 + \left|x\right| \cdot 0.3275911}\\ \end{array} \]

Alternative 10: 98.4% accurate, 1.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + \left|x\right| \cdot 0.3275911}\\ t_1 := 1 + x \cdot 0.3275911\\ \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.029667143 + x \cdot -0.2193742730720041}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(e^{x \cdot \left(-x\right)} \cdot \left(0.254829592 + t_0 \cdot \left(-0.284496736 + t_0 \cdot \left(1.421413741 + \frac{1}{t_1} \cdot \left(-1.453152027 + \frac{1.061405429}{t_1}\right)\right)\right)\right)\right) \cdot \frac{-1}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* (fabs x) 0.3275911))))
        (t_1 (+ 1.0 (* x 0.3275911))))
   (if (<= (fabs x) 5e-12)
     (/
      (+ 1.999999999e-9 (* x 2.256772713883663))
      (+
       1.0
       (/
        (+
         0.254829592
         (/
          (+
           -0.284496736
           (/ (+ 1.029667143 (* x -0.2193742730720041)) (fma x 0.3275911 1.0)))
          (fma x 0.3275911 1.0)))
        (* (fma x 0.3275911 1.0) (exp (* x x))))))
     (+
      1.0
      (*
       (*
        (exp (* x (- x)))
        (+
         0.254829592
         (*
          t_0
          (+
           -0.284496736
           (*
            t_0
            (+
             1.421413741
             (* (/ 1.0 t_1) (+ -1.453152027 (/ 1.061405429 t_1)))))))))
       (/ -1.0 t_1))))))

C

double code(double x) {
	double t_0 = 1.0 / (1.0 + (fabs(x) * 0.3275911));
	double t_1 = 1.0 + (x * 0.3275911);
	double tmp;
	if (fabs(x) <= 5e-12) {
		tmp = (1.999999999e-9 + (x * 2.256772713883663)) / (1.0 + ((0.254829592 + ((-0.284496736 + ((1.029667143 + (x * -0.2193742730720041)) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / (fma(x, 0.3275911, 1.0) * exp((x * x)))));
	} else {
		tmp = 1.0 + ((exp((x * -x)) * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + ((1.0 / t_1) * (-1.453152027 + (1.061405429 / t_1))))))))) * (-1.0 / t_1));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(abs(x) * 0.3275911)))
	t_1 = Float64(1.0 + Float64(x * 0.3275911))
	tmp = 0.0
	if (abs(x) <= 5e-12)
		tmp = Float64(Float64(1.999999999e-9 + Float64(x * 2.256772713883663)) / Float64(1.0 + Float64(Float64(0.254829592 + Float64(Float64(-0.284496736 + Float64(Float64(1.029667143 + Float64(x * -0.2193742730720041)) / fma(x, 0.3275911, 1.0))) / fma(x, 0.3275911, 1.0))) / Float64(fma(x, 0.3275911, 1.0) * exp(Float64(x * x))))));
	else
		tmp = Float64(1.0 + Float64(Float64(exp(Float64(x * Float64(-x))) * Float64(0.254829592 + Float64(t_0 * Float64(-0.284496736 + Float64(t_0 * Float64(1.421413741 + Float64(Float64(1.0 / t_1) * Float64(-1.453152027 + Float64(1.061405429 / t_1))))))))) * Float64(-1.0 / t_1)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 5e-12], N[(N[(1.999999999e-9 + N[(x * 2.256772713883663), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.254829592 + N[(N[(-0.284496736 + N[(N[(1.029667143 + N[(x * -0.2193742730720041), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(x * 0.3275911 + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * 0.3275911 + 1.0), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] * N[(0.254829592 + N[(t$95$0 * N[(-0.284496736 + N[(t$95$0 * N[(1.421413741 + N[(N[(1.0 / t$95$1), $MachinePrecision] * N[(-1.453152027 + N[(1.061405429 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 4.9999999999999997e-12

   1. Initial program 57.7%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Applied egg-rr 57.7%

      \[\leadsto \color{blue}{\frac{1 - \left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}\right) \cdot \left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}}} \]

   5. Simplified 57.7%

      \[\leadsto \color{blue}{\frac{1 + \frac{\frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{e^{x \cdot x}} \cdot \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}} \]

   6. Taylor expanded in x around 0 99.6%

      \[\leadsto \frac{\color{blue}{1.999999999 \cdot 10^{-9} + 2.256772713883663 \cdot x}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]
   7. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \frac{1.999999999 \cdot 10^{-9} + \color{blue}{x \cdot 2.256772713883663}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   8. Simplified 99.6%

      \[\leadsto \frac{\color{blue}{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   9. Taylor expanded in x around 0 99.6%

      \[\leadsto \frac{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{\color{blue}{1.029667143 + -0.2193742730720041 \cdot x}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]
   10. Step-by-step derivation

       1. *-commutative 99.6%

          \[\leadsto \frac{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.029667143 + \color{blue}{x \cdot -0.2193742730720041}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   11. Simplified 99.6%

       \[\leadsto \frac{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{\color{blue}{1.029667143 + x \cdot -0.2193742730720041}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   if 4.9999999999999997e-12 < (fabs.f64 x)

   1. Initial program 99.1%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
   2. Step-by-step derivation

      1. associate-*l* 99.1%

         \[\leadsto 1 - \color{blue}{\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]
   4. Step-by-step derivation

      1. pow1 99.1%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   5. Applied egg-rr 99.1%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   6. Step-by-step derivation

      1. unpow1 99.1%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      2. *-commutative 99.1%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left|x\right| \cdot 0.3275911}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      3. unpow1 99.1%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left|\color{blue}{{x}^{1}}\right| \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      4. sqr-pow 54.5%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      5. fabs-sqr 54.5%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)} \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      6. sqr-pow 97.4%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{x}^{1}} \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      7. unpow1 97.4%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{x} \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   7. Simplified 97.4%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{x \cdot 0.3275911}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   8. Step-by-step derivation

      1. pow1 99.1%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   9. Applied egg-rr 97.4%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   10. Step-by-step derivation

       1. unpow1 99.1%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       2. *-commutative 99.1%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left|x\right| \cdot 0.3275911}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       3. unpow1 99.1%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left|\color{blue}{{x}^{1}}\right| \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. sqr-pow 54.5%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       5. fabs-sqr 54.5%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)} \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       6. sqr-pow 97.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{x}^{1}} \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       7. unpow1 97.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{x} \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   11. Simplified 97.4%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{x \cdot 0.3275911}} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   12. Step-by-step derivation

       1. pow1 99.1%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   13. Applied egg-rr 97.4%

       \[\leadsto 1 - \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   14. Step-by-step derivation

       1. unpow1 99.1%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       2. *-commutative 99.1%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left|x\right| \cdot 0.3275911}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       3. unpow1 99.1%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left|\color{blue}{{x}^{1}}\right| \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. sqr-pow 54.5%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       5. fabs-sqr 54.5%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)} \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       6. sqr-pow 97.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{x}^{1}} \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       7. unpow1 97.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{x} \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   15. Simplified 97.3%

       \[\leadsto 1 - \frac{1}{1 + \color{blue}{x \cdot 0.3275911}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.029667143 + x \cdot -0.2193742730720041}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(e^{x \cdot \left(-x\right)} \cdot \left(0.254829592 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right)\right)\right)\right) \cdot \frac{-1}{1 + x \cdot 0.3275911}\\ \end{array} \]

Alternative 11: 98.4% accurate, 1.9× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := \frac{1}{1 + \left|x\right| \cdot 0.3275911}\\ t_1 := 1 + x \cdot 0.3275911\\ \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}{1 + \frac{0.254829592 + \left(0.745170407 + x \cdot -0.8007952583978091\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(e^{x \cdot \left(-x\right)} \cdot \left(0.254829592 + t_0 \cdot \left(-0.284496736 + t_0 \cdot \left(1.421413741 + \frac{1}{t_1} \cdot \left(-1.453152027 + \frac{1.061405429}{t_1}\right)\right)\right)\right)\right) \cdot \frac{-1}{t_1}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (/ 1.0 (+ 1.0 (* (fabs x) 0.3275911))))
        (t_1 (+ 1.0 (* x 0.3275911))))
   (if (<= (fabs x) 5e-12)
     (/
      (+ 1.999999999e-9 (* x 2.256772713883663))
      (+
       1.0
       (/
        (+ 0.254829592 (+ 0.745170407 (* x -0.8007952583978091)))
        (* (fma x 0.3275911 1.0) (exp (* x x))))))
     (+
      1.0
      (*
       (*
        (exp (* x (- x)))
        (+
         0.254829592
         (*
          t_0
          (+
           -0.284496736
           (*
            t_0
            (+
             1.421413741
             (* (/ 1.0 t_1) (+ -1.453152027 (/ 1.061405429 t_1)))))))))
       (/ -1.0 t_1))))))

C

double code(double x) {
	double t_0 = 1.0 / (1.0 + (fabs(x) * 0.3275911));
	double t_1 = 1.0 + (x * 0.3275911);
	double tmp;
	if (fabs(x) <= 5e-12) {
		tmp = (1.999999999e-9 + (x * 2.256772713883663)) / (1.0 + ((0.254829592 + (0.745170407 + (x * -0.8007952583978091))) / (fma(x, 0.3275911, 1.0) * exp((x * x)))));
	} else {
		tmp = 1.0 + ((exp((x * -x)) * (0.254829592 + (t_0 * (-0.284496736 + (t_0 * (1.421413741 + ((1.0 / t_1) * (-1.453152027 + (1.061405429 / t_1))))))))) * (-1.0 / t_1));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = Float64(1.0 / Float64(1.0 + Float64(abs(x) * 0.3275911)))
	t_1 = Float64(1.0 + Float64(x * 0.3275911))
	tmp = 0.0
	if (abs(x) <= 5e-12)
		tmp = Float64(Float64(1.999999999e-9 + Float64(x * 2.256772713883663)) / Float64(1.0 + Float64(Float64(0.254829592 + Float64(0.745170407 + Float64(x * -0.8007952583978091))) / Float64(fma(x, 0.3275911, 1.0) * exp(Float64(x * x))))));
	else
		tmp = Float64(1.0 + Float64(Float64(exp(Float64(x * Float64(-x))) * Float64(0.254829592 + Float64(t_0 * Float64(-0.284496736 + Float64(t_0 * Float64(1.421413741 + Float64(Float64(1.0 / t_1) * Float64(-1.453152027 + Float64(1.061405429 / t_1))))))))) * Float64(-1.0 / t_1)));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[(1.0 / N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$1 = N[(1.0 + N[(x * 0.3275911), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 5e-12], N[(N[(1.999999999e-9 + N[(x * 2.256772713883663), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.254829592 + N[(0.745170407 + N[(x * -0.8007952583978091), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * 0.3275911 + 1.0), $MachinePrecision] * N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 + N[(N[(N[Exp[N[(x * (-x)), $MachinePrecision]], $MachinePrecision] * N[(0.254829592 + N[(t$95$0 * N[(-0.284496736 + N[(t$95$0 * N[(1.421413741 + N[(N[(1.0 / t$95$1), $MachinePrecision] * N[(-1.453152027 + N[(1.061405429 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(-1.0 / t$95$1), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 4.9999999999999997e-12

   1. Initial program 57.7%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Applied egg-rr 57.7%

      \[\leadsto \color{blue}{\frac{1 - \left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}\right) \cdot \left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}}} \]

   5. Simplified 57.7%

      \[\leadsto \color{blue}{\frac{1 + \frac{\frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{e^{x \cdot x}} \cdot \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}} \]

   6. Taylor expanded in x around 0 99.6%

      \[\leadsto \frac{\color{blue}{1.999999999 \cdot 10^{-9} + 2.256772713883663 \cdot x}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]
   7. Step-by-step derivation

      1. *-commutative 99.6%

         \[\leadsto \frac{1.999999999 \cdot 10^{-9} + \color{blue}{x \cdot 2.256772713883663}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   8. Simplified 99.6%

      \[\leadsto \frac{\color{blue}{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   9. Taylor expanded in x around 0 99.6%

      \[\leadsto \frac{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}{1 + \frac{0.254829592 + \color{blue}{\left(0.745170407 + -0.8007952583978091 \cdot x\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]
   10. Step-by-step derivation

       1. *-commutative 99.6%

          \[\leadsto \frac{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}{1 + \frac{0.254829592 + \left(0.745170407 + \color{blue}{x \cdot -0.8007952583978091}\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   11. Simplified 99.6%

       \[\leadsto \frac{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}{1 + \frac{0.254829592 + \color{blue}{\left(0.745170407 + x \cdot -0.8007952583978091\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   if 4.9999999999999997e-12 < (fabs.f64 x)

   1. Initial program 99.1%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]
   2. Step-by-step derivation

      1. associate-*l* 99.1%

         \[\leadsto 1 - \color{blue}{\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|}\right)} \]

   3. Simplified 99.1%

      \[\leadsto \color{blue}{1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + 0.3275911 \cdot \left|x\right|}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right)} \]
   4. Step-by-step derivation

      1. pow1 99.1%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   5. Applied egg-rr 99.1%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   6. Step-by-step derivation

      1. unpow1 99.1%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      2. *-commutative 99.1%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left|x\right| \cdot 0.3275911}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      3. unpow1 99.1%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left|\color{blue}{{x}^{1}}\right| \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      4. sqr-pow 54.5%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      5. fabs-sqr 54.5%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)} \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      6. sqr-pow 97.4%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{x}^{1}} \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

      7. unpow1 97.4%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{x} \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   7. Simplified 97.4%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{x \cdot 0.3275911}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   8. Step-by-step derivation

      1. pow1 99.1%

         \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   9. Applied egg-rr 97.4%

      \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   10. Step-by-step derivation

       1. unpow1 99.1%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       2. *-commutative 99.1%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left|x\right| \cdot 0.3275911}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       3. unpow1 99.1%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left|\color{blue}{{x}^{1}}\right| \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. sqr-pow 54.5%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       5. fabs-sqr 54.5%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)} \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       6. sqr-pow 97.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{x}^{1}} \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       7. unpow1 97.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{x} \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   11. Simplified 97.4%

       \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + \color{blue}{x \cdot 0.3275911}} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   12. Step-by-step derivation

       1. pow1 99.1%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   13. Applied egg-rr 97.4%

       \[\leadsto 1 - \frac{1}{1 + \color{blue}{{\left(0.3275911 \cdot \left|x\right|\right)}^{1}}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
   14. Step-by-step derivation

       1. unpow1 99.1%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{0.3275911 \cdot \left|x\right|}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       2. *-commutative 99.1%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left|x\right| \cdot 0.3275911}}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       3. unpow1 99.1%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left|\color{blue}{{x}^{1}}\right| \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       4. sqr-pow 54.5%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right| \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       5. fabs-sqr 54.5%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{\left({x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}\right)} \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       6. sqr-pow 97.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{{x}^{1}} \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

       7. unpow1 97.4%

          \[\leadsto 1 - \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + \color{blue}{x} \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]

   15. Simplified 97.3%

       \[\leadsto 1 - \frac{1}{1 + \color{blue}{x \cdot 0.3275911}} \cdot \left(\left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right)\right)\right) \cdot e^{-x \cdot x}\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.4%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 5 \cdot 10^{-12}:\\ \;\;\;\;\frac{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}{1 + \frac{0.254829592 + \left(0.745170407 + x \cdot -0.8007952583978091\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;1 + \left(e^{x \cdot \left(-x\right)} \cdot \left(0.254829592 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(-0.284496736 + \frac{1}{1 + \left|x\right| \cdot 0.3275911} \cdot \left(1.421413741 + \frac{1}{1 + x \cdot 0.3275911} \cdot \left(-1.453152027 + \frac{1.061405429}{1 + x \cdot 0.3275911}\right)\right)\right)\right)\right) \cdot \frac{-1}{1 + x \cdot 0.3275911}\\ \end{array} \]

Alternative 12: 98.0% accurate, 2.6× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x \cdot x}\\ \mathbf{if}\;\left|x\right| \leq 0.01:\\ \;\;\;\;\frac{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}{1 + \frac{0.254829592 + \left(0.745170407 + x \cdot -0.8007952583978091\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot t_0}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.254829592}{t_0 \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (* x x))))
   (if (<= (fabs x) 0.01)
     (/
      (+ 1.999999999e-9 (* x 2.256772713883663))
      (+
       1.0
       (/
        (+ 0.254829592 (+ 0.745170407 (* x -0.8007952583978091)))
        (* (fma x 0.3275911 1.0) t_0))))
     (- 1.0 (/ 0.254829592 (* t_0 (+ 1.0 (* (fabs x) 0.3275911))))))))

C

double code(double x) {
	double t_0 = exp((x * x));
	double tmp;
	if (fabs(x) <= 0.01) {
		tmp = (1.999999999e-9 + (x * 2.256772713883663)) / (1.0 + ((0.254829592 + (0.745170407 + (x * -0.8007952583978091))) / (fma(x, 0.3275911, 1.0) * t_0)));
	} else {
		tmp = 1.0 - (0.254829592 / (t_0 * (1.0 + (fabs(x) * 0.3275911))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(x * x))
	tmp = 0.0
	if (abs(x) <= 0.01)
		tmp = Float64(Float64(1.999999999e-9 + Float64(x * 2.256772713883663)) / Float64(1.0 + Float64(Float64(0.254829592 + Float64(0.745170407 + Float64(x * -0.8007952583978091))) / Float64(fma(x, 0.3275911, 1.0) * t_0))));
	else
		tmp = Float64(1.0 - Float64(0.254829592 / Float64(t_0 * Float64(1.0 + Float64(abs(x) * 0.3275911)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.01], N[(N[(1.999999999e-9 + N[(x * 2.256772713883663), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(N[(0.254829592 + N[(0.745170407 + N[(x * -0.8007952583978091), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(x * 0.3275911 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.254829592 / N[(t$95$0 * N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0100000000000000002

   1. Initial program 58.0%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Applied egg-rr 58.1%

      \[\leadsto \color{blue}{\frac{1 - \left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}\right) \cdot \left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}}} \]

   5. Simplified 57.3%

      \[\leadsto \color{blue}{\frac{1 + \frac{\frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{e^{x \cdot x}} \cdot \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}} \]

   6. Taylor expanded in x around 0 97.1%

      \[\leadsto \frac{\color{blue}{1.999999999 \cdot 10^{-9} + 2.256772713883663 \cdot x}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]
   7. Step-by-step derivation

      1. *-commutative 97.1%

         \[\leadsto \frac{1.999999999 \cdot 10^{-9} + \color{blue}{x \cdot 2.256772713883663}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   8. Simplified 97.1%

      \[\leadsto \frac{\color{blue}{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   9. Taylor expanded in x around 0 97.1%

      \[\leadsto \frac{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}{1 + \frac{0.254829592 + \color{blue}{\left(0.745170407 + -0.8007952583978091 \cdot x\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]
   10. Step-by-step derivation

       1. *-commutative 97.1%

          \[\leadsto \frac{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}{1 + \frac{0.254829592 + \left(0.745170407 + \color{blue}{x \cdot -0.8007952583978091}\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   11. Simplified 97.1%

       \[\leadsto \frac{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}{1 + \frac{0.254829592 + \color{blue}{\left(0.745170407 + x \cdot -0.8007952583978091\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   if 0.0100000000000000002 < (fabs.f64 x)

   1. Initial program 100.0%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   5. Applied egg-rr 100.0%

      \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\left(\frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot -0.284496736 + \frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]
   6. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot \left(-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]

      2. fma-def 100.0%

         \[\leadsto 1 - \frac{0.254829592 + \frac{1}{\color{blue}{0.3275911 \cdot \left|x\right| + 1}} \cdot \left(-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]

      3. associate-*l/ 100.0%

         \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\frac{1 \cdot \left(-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{0.3275911 \cdot \left|x\right| + 1}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]

   7. Simplified 98.6%

      \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]

   8. Taylor expanded in x around inf 98.8%

      \[\leadsto \color{blue}{1 - 0.254829592 \cdot \frac{1}{\left(0.3275911 \cdot \left|x\right| + 1\right) \cdot e^{{x}^{2}}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 98.8%

         \[\leadsto 1 - \color{blue}{\frac{0.254829592 \cdot 1}{\left(0.3275911 \cdot \left|x\right| + 1\right) \cdot e^{{x}^{2}}}} \]

      2. metadata-eval 98.8%

         \[\leadsto 1 - \frac{\color{blue}{0.254829592}}{\left(0.3275911 \cdot \left|x\right| + 1\right) \cdot e^{{x}^{2}}} \]

      3. fma-def 98.8%

         \[\leadsto 1 - \frac{0.254829592}{\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{{x}^{2}}} \]

      4. unpow2 98.8%

         \[\leadsto 1 - \frac{0.254829592}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot e^{\color{blue}{x \cdot x}}} \]

   10. Simplified 98.8%

       \[\leadsto \color{blue}{1 - \frac{0.254829592}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot e^{x \cdot x}}} \]

   11. Taylor expanded in x around 0 98.8%

       \[\leadsto 1 - \frac{0.254829592}{\color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)} \cdot e^{x \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 98.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.01:\\ \;\;\;\;\frac{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}{1 + \frac{0.254829592 + \left(0.745170407 + x \cdot -0.8007952583978091\right)}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.254829592}{e^{x \cdot x} \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)}\\ \end{array} \]

Alternative 13: 98.0% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} t_0 := e^{x \cdot x}\\ \mathbf{if}\;\left|x\right| \leq 0.01:\\ \;\;\;\;\frac{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}{1 + \frac{0.999999999}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot t_0}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.254829592}{t_0 \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (let* ((t_0 (exp (* x x))))
   (if (<= (fabs x) 0.01)
     (/
      (+ 1.999999999e-9 (* x 2.256772713883663))
      (+ 1.0 (/ 0.999999999 (* (fma x 0.3275911 1.0) t_0))))
     (- 1.0 (/ 0.254829592 (* t_0 (+ 1.0 (* (fabs x) 0.3275911))))))))

C

double code(double x) {
	double t_0 = exp((x * x));
	double tmp;
	if (fabs(x) <= 0.01) {
		tmp = (1.999999999e-9 + (x * 2.256772713883663)) / (1.0 + (0.999999999 / (fma(x, 0.3275911, 1.0) * t_0)));
	} else {
		tmp = 1.0 - (0.254829592 / (t_0 * (1.0 + (fabs(x) * 0.3275911))));
	}
	return tmp;
}

Julia

function code(x)
	t_0 = exp(Float64(x * x))
	tmp = 0.0
	if (abs(x) <= 0.01)
		tmp = Float64(Float64(1.999999999e-9 + Float64(x * 2.256772713883663)) / Float64(1.0 + Float64(0.999999999 / Float64(fma(x, 0.3275911, 1.0) * t_0))));
	else
		tmp = Float64(1.0 - Float64(0.254829592 / Float64(t_0 * Float64(1.0 + Float64(abs(x) * 0.3275911)))));
	end
	return tmp
end

Wolfram

code[x_] := Block[{t$95$0 = N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision]}, If[LessEqual[N[Abs[x], $MachinePrecision], 0.01], N[(N[(1.999999999e-9 + N[(x * 2.256772713883663), $MachinePrecision]), $MachinePrecision] / N[(1.0 + N[(0.999999999 / N[(N[(x * 0.3275911 + 1.0), $MachinePrecision] * t$95$0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(1.0 - N[(0.254829592 / N[(t$95$0 * N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 0.0100000000000000002

   1. Initial program 58.0%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Applied egg-rr 58.1%

      \[\leadsto \color{blue}{\frac{1 - \left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}\right) \cdot \left(\frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}\right)}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{-x \cdot x}}} \]

   5. Simplified 57.3%

      \[\leadsto \color{blue}{\frac{1 + \frac{\frac{-0.254829592 - \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{e^{x \cdot x}} \cdot \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}} \]

   6. Taylor expanded in x around 0 97.1%

      \[\leadsto \frac{\color{blue}{1.999999999 \cdot 10^{-9} + 2.256772713883663 \cdot x}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]
   7. Step-by-step derivation

      1. *-commutative 97.1%

         \[\leadsto \frac{1.999999999 \cdot 10^{-9} + \color{blue}{x \cdot 2.256772713883663}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   8. Simplified 97.1%

      \[\leadsto \frac{\color{blue}{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}}{1 + \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   9. Taylor expanded in x around 0 97.0%

      \[\leadsto \frac{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}{1 + \frac{0.254829592 + \color{blue}{0.745170407}}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}} \]

   if 0.0100000000000000002 < (fabs.f64 x)

   1. Initial program 100.0%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 100.0%

      \[\leadsto \color{blue}{1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   5. Applied egg-rr 100.0%

      \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\left(\frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot -0.284496736 + \frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]
   6. Step-by-step derivation

      1. distribute-lft-in 100.0%

         \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot \left(-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]

      2. fma-def 100.0%

         \[\leadsto 1 - \frac{0.254829592 + \frac{1}{\color{blue}{0.3275911 \cdot \left|x\right| + 1}} \cdot \left(-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]

      3. associate-*l/ 100.0%

         \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\frac{1 \cdot \left(-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{0.3275911 \cdot \left|x\right| + 1}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]

   7. Simplified 98.6%

      \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]

   8. Taylor expanded in x around inf 98.8%

      \[\leadsto \color{blue}{1 - 0.254829592 \cdot \frac{1}{\left(0.3275911 \cdot \left|x\right| + 1\right) \cdot e^{{x}^{2}}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 98.8%

         \[\leadsto 1 - \color{blue}{\frac{0.254829592 \cdot 1}{\left(0.3275911 \cdot \left|x\right| + 1\right) \cdot e^{{x}^{2}}}} \]

      2. metadata-eval 98.8%

         \[\leadsto 1 - \frac{\color{blue}{0.254829592}}{\left(0.3275911 \cdot \left|x\right| + 1\right) \cdot e^{{x}^{2}}} \]

      3. fma-def 98.8%

         \[\leadsto 1 - \frac{0.254829592}{\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{{x}^{2}}} \]

      4. unpow2 98.8%

         \[\leadsto 1 - \frac{0.254829592}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot e^{\color{blue}{x \cdot x}}} \]

   10. Simplified 98.8%

       \[\leadsto \color{blue}{1 - \frac{0.254829592}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot e^{x \cdot x}}} \]

   11. Taylor expanded in x around 0 98.8%

       \[\leadsto 1 - \frac{0.254829592}{\color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)} \cdot e^{x \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.9%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 0.01:\\ \;\;\;\;\frac{1.999999999 \cdot 10^{-9} + x \cdot 2.256772713883663}{1 + \frac{0.999999999}{\mathsf{fma}\left(x, 0.3275911, 1\right) \cdot e^{x \cdot x}}}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.254829592}{e^{x \cdot x} \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)}\\ \end{array} \]

Alternative 14: 97.4% accurate, 2.7× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-5}:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.254829592}{e^{x \cdot x} \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)}\\ \end{array} \end{array} \]

FPCore

(FPCore (x)
 :precision binary64
 (if (<= (fabs x) 1e-5)
   1e-9
   (- 1.0 (/ 0.254829592 (* (exp (* x x)) (+ 1.0 (* (fabs x) 0.3275911)))))))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 1e-5) {
		tmp = 1e-9;
	} else {
		tmp = 1.0 - (0.254829592 / (exp((x * x)) * (1.0 + (fabs(x) * 0.3275911))));
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (abs(x) <= 1d-5) then
        tmp = 1d-9
    else
        tmp = 1.0d0 - (0.254829592d0 / (exp((x * x)) * (1.0d0 + (abs(x) * 0.3275911d0))))
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 1e-5) {
		tmp = 1e-9;
	} else {
		tmp = 1.0 - (0.254829592 / (Math.exp((x * x)) * (1.0 + (Math.abs(x) * 0.3275911))));
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 1e-5:
		tmp = 1e-9
	else:
		tmp = 1.0 - (0.254829592 / (math.exp((x * x)) * (1.0 + (math.fabs(x) * 0.3275911))))
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 1e-5)
		tmp = 1e-9;
	else
		tmp = Float64(1.0 - Float64(0.254829592 / Float64(exp(Float64(x * x)) * Float64(1.0 + Float64(abs(x) * 0.3275911)))));
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 1e-5)
		tmp = 1e-9;
	else
		tmp = 1.0 - (0.254829592 / (exp((x * x)) * (1.0 + (abs(x) * 0.3275911))));
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 1e-5], 1e-9, N[(1.0 - N[(0.254829592 / N[(N[Exp[N[(x * x), $MachinePrecision]], $MachinePrecision] * N[(1.0 + N[(N[Abs[x], $MachinePrecision] * 0.3275911), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000008e-5

   1. Initial program 57.8%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 57.8%

      \[\leadsto \color{blue}{1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   5. Applied egg-rr 57.8%

      \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\left(\frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot -0.284496736 + \frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]
   6. Step-by-step derivation

      1. distribute-lft-in 57.8%

         \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot \left(-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]

      2. fma-def 57.8%

         \[\leadsto 1 - \frac{0.254829592 + \frac{1}{\color{blue}{0.3275911 \cdot \left|x\right| + 1}} \cdot \left(-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]

      3. associate-*l/ 57.8%

         \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\frac{1 \cdot \left(-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{0.3275911 \cdot \left|x\right| + 1}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]

   7. Simplified 57.0%

      \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]

   8. Taylor expanded in x around 0 54.1%

      \[\leadsto \color{blue}{\left(1 + 1.737648763 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right) - 2.737648762 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}} \]
   9. Step-by-step derivation

      1. associate--l+ 53.0%

         \[\leadsto \color{blue}{1 + \left(1.737648763 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1} - 2.737648762 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right)} \]

      2. +-commutative 53.0%

         \[\leadsto 1 + \left(1.737648763 \cdot \frac{1}{\color{blue}{1 + 0.3275911 \cdot \left|x\right|}} - 2.737648762 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right) \]

      3. +-commutative 53.0%

         \[\leadsto 1 + \left(1.737648763 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} - 2.737648762 \cdot \frac{1}{\color{blue}{1 + 0.3275911 \cdot \left|x\right|}}\right) \]

      4. distribute-rgt-out-- 53.0%

         \[\leadsto 1 + \color{blue}{\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.737648763 - 2.737648762\right)} \]

      5. +-commutative 53.0%

         \[\leadsto 1 + \frac{1}{\color{blue}{0.3275911 \cdot \left|x\right| + 1}} \cdot \left(1.737648763 - 2.737648762\right) \]

      6. fma-def 53.0%

         \[\leadsto 1 + \frac{1}{\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}} \cdot \left(1.737648763 - 2.737648762\right) \]

      7. unpow1 53.0%

         \[\leadsto 1 + \frac{1}{\mathsf{fma}\left(0.3275911, \left|\color{blue}{{x}^{1}}\right|, 1\right)} \cdot \left(1.737648763 - 2.737648762\right) \]

      8. sqr-pow 27.4%

         \[\leadsto 1 + \frac{1}{\mathsf{fma}\left(0.3275911, \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|, 1\right)} \cdot \left(1.737648763 - 2.737648762\right) \]

      9. fabs-sqr 27.4%

         \[\leadsto 1 + \frac{1}{\mathsf{fma}\left(0.3275911, \color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}, 1\right)} \cdot \left(1.737648763 - 2.737648762\right) \]

      10. sqr-pow 52.8%

          \[\leadsto 1 + \frac{1}{\mathsf{fma}\left(0.3275911, \color{blue}{{x}^{1}}, 1\right)} \cdot \left(1.737648763 - 2.737648762\right) \]

      11. unpow1 52.8%

          \[\leadsto 1 + \frac{1}{\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right)} \cdot \left(1.737648763 - 2.737648762\right) \]

      12. metadata-eval 56.3%

          \[\leadsto 1 + \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot \color{blue}{-0.999999999} \]

   10. Simplified 56.3%

       \[\leadsto \color{blue}{1 + \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot -0.999999999} \]

   11. Taylor expanded in x around 0 96.4%

       \[\leadsto \color{blue}{10^{-9}} \]

   if 1.00000000000000008e-5 < (fabs.f64 x)

   1. Initial program 99.9%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   5. Applied egg-rr 99.9%

      \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\left(\frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot -0.284496736 + \frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]
   6. Step-by-step derivation

      1. distribute-lft-in 99.9%

         \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot \left(-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]

      2. fma-def 99.9%

         \[\leadsto 1 - \frac{0.254829592 + \frac{1}{\color{blue}{0.3275911 \cdot \left|x\right| + 1}} \cdot \left(-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]

      3. associate-*l/ 99.9%

         \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\frac{1 \cdot \left(-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{0.3275911 \cdot \left|x\right| + 1}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]

   7. Simplified 98.5%

      \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]

   8. Taylor expanded in x around inf 98.2%

      \[\leadsto \color{blue}{1 - 0.254829592 \cdot \frac{1}{\left(0.3275911 \cdot \left|x\right| + 1\right) \cdot e^{{x}^{2}}}} \]
   9. Step-by-step derivation

      1. associate-*r/ 98.2%

         \[\leadsto 1 - \color{blue}{\frac{0.254829592 \cdot 1}{\left(0.3275911 \cdot \left|x\right| + 1\right) \cdot e^{{x}^{2}}}} \]

      2. metadata-eval 98.2%

         \[\leadsto 1 - \frac{\color{blue}{0.254829592}}{\left(0.3275911 \cdot \left|x\right| + 1\right) \cdot e^{{x}^{2}}} \]

      3. fma-def 98.2%

         \[\leadsto 1 - \frac{0.254829592}{\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot e^{{x}^{2}}} \]

      4. unpow2 98.2%

         \[\leadsto 1 - \frac{0.254829592}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot e^{\color{blue}{x \cdot x}}} \]

   10. Simplified 98.2%

       \[\leadsto \color{blue}{1 - \frac{0.254829592}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot e^{x \cdot x}}} \]

   11. Taylor expanded in x around 0 98.2%

       \[\leadsto 1 - \frac{0.254829592}{\color{blue}{\left(0.3275911 \cdot \left|x\right| + 1\right)} \cdot e^{x \cdot x}} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-5}:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1 - \frac{0.254829592}{e^{x \cdot x} \cdot \left(1 + \left|x\right| \cdot 0.3275911\right)}\\ \end{array} \]

Alternative 15: 97.4% accurate, 8.3× speedup

Math

\[\begin{array}{l} \\ \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-5}:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \end{array} \]

FPCore

(FPCore (x) :precision binary64 (if (<= (fabs x) 1e-5) 1e-9 1.0))

C

double code(double x) {
	double tmp;
	if (fabs(x) <= 1e-5) {
		tmp = 1e-9;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    real(8) :: tmp
    if (abs(x) <= 1d-5) then
        tmp = 1d-9
    else
        tmp = 1.0d0
    end if
    code = tmp
end function

Java

public static double code(double x) {
	double tmp;
	if (Math.abs(x) <= 1e-5) {
		tmp = 1e-9;
	} else {
		tmp = 1.0;
	}
	return tmp;
}

Python

def code(x):
	tmp = 0
	if math.fabs(x) <= 1e-5:
		tmp = 1e-9
	else:
		tmp = 1.0
	return tmp

Julia

function code(x)
	tmp = 0.0
	if (abs(x) <= 1e-5)
		tmp = 1e-9;
	else
		tmp = 1.0;
	end
	return tmp
end

MATLAB

function tmp_2 = code(x)
	tmp = 0.0;
	if (abs(x) <= 1e-5)
		tmp = 1e-9;
	else
		tmp = 1.0;
	end
	tmp_2 = tmp;
end

Wolfram

code[x_] := If[LessEqual[N[Abs[x], $MachinePrecision], 1e-5], 1e-9, 1.0]

Derivation

1. Split input into 2 regimes

2. if (fabs.f64 x) < 1.00000000000000008e-5

   1. Initial program 57.8%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 57.8%

      \[\leadsto \color{blue}{1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   5. Applied egg-rr 57.8%

      \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\left(\frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot -0.284496736 + \frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]
   6. Step-by-step derivation

      1. distribute-lft-in 57.8%

         \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot \left(-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]

      2. fma-def 57.8%

         \[\leadsto 1 - \frac{0.254829592 + \frac{1}{\color{blue}{0.3275911 \cdot \left|x\right| + 1}} \cdot \left(-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]

      3. associate-*l/ 57.8%

         \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\frac{1 \cdot \left(-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{0.3275911 \cdot \left|x\right| + 1}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]

   7. Simplified 57.0%

      \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]

   8. Taylor expanded in x around 0 54.1%

      \[\leadsto \color{blue}{\left(1 + 1.737648763 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right) - 2.737648762 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}} \]
   9. Step-by-step derivation

      1. associate--l+ 53.0%

         \[\leadsto \color{blue}{1 + \left(1.737648763 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1} - 2.737648762 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right)} \]

      2. +-commutative 53.0%

         \[\leadsto 1 + \left(1.737648763 \cdot \frac{1}{\color{blue}{1 + 0.3275911 \cdot \left|x\right|}} - 2.737648762 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right) \]

      3. +-commutative 53.0%

         \[\leadsto 1 + \left(1.737648763 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} - 2.737648762 \cdot \frac{1}{\color{blue}{1 + 0.3275911 \cdot \left|x\right|}}\right) \]

      4. distribute-rgt-out-- 53.0%

         \[\leadsto 1 + \color{blue}{\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.737648763 - 2.737648762\right)} \]

      5. +-commutative 53.0%

         \[\leadsto 1 + \frac{1}{\color{blue}{0.3275911 \cdot \left|x\right| + 1}} \cdot \left(1.737648763 - 2.737648762\right) \]

      6. fma-def 53.0%

         \[\leadsto 1 + \frac{1}{\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}} \cdot \left(1.737648763 - 2.737648762\right) \]

      7. unpow1 53.0%

         \[\leadsto 1 + \frac{1}{\mathsf{fma}\left(0.3275911, \left|\color{blue}{{x}^{1}}\right|, 1\right)} \cdot \left(1.737648763 - 2.737648762\right) \]

      8. sqr-pow 27.4%

         \[\leadsto 1 + \frac{1}{\mathsf{fma}\left(0.3275911, \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|, 1\right)} \cdot \left(1.737648763 - 2.737648762\right) \]

      9. fabs-sqr 27.4%

         \[\leadsto 1 + \frac{1}{\mathsf{fma}\left(0.3275911, \color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}, 1\right)} \cdot \left(1.737648763 - 2.737648762\right) \]

      10. sqr-pow 52.8%

          \[\leadsto 1 + \frac{1}{\mathsf{fma}\left(0.3275911, \color{blue}{{x}^{1}}, 1\right)} \cdot \left(1.737648763 - 2.737648762\right) \]

      11. unpow1 52.8%

          \[\leadsto 1 + \frac{1}{\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right)} \cdot \left(1.737648763 - 2.737648762\right) \]

      12. metadata-eval 56.3%

          \[\leadsto 1 + \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot \color{blue}{-0.999999999} \]

   10. Simplified 56.3%

       \[\leadsto \color{blue}{1 + \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot -0.999999999} \]

   11. Taylor expanded in x around 0 96.4%

       \[\leadsto \color{blue}{10^{-9}} \]

   if 1.00000000000000008e-5 < (fabs.f64 x)

   1. Initial program 99.9%

      \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

   3. Simplified 99.9%

      \[\leadsto \color{blue}{1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

   5. Applied egg-rr 99.9%

      \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\left(\frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot -0.284496736 + \frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]
   6. Step-by-step derivation

      1. distribute-lft-in 99.9%

         \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot \left(-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]

      2. fma-def 99.9%

         \[\leadsto 1 - \frac{0.254829592 + \frac{1}{\color{blue}{0.3275911 \cdot \left|x\right| + 1}} \cdot \left(-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]

      3. associate-*l/ 99.9%

         \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\frac{1 \cdot \left(-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{0.3275911 \cdot \left|x\right| + 1}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]

   7. Simplified 98.5%

      \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]

   8. Taylor expanded in x around 0 95.2%

      \[\leadsto \color{blue}{\left(1 + 1.737648763 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right) - 2.737648762 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}} \]
   9. Step-by-step derivation

      1. associate--l+ 95.2%

         \[\leadsto \color{blue}{1 + \left(1.737648763 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1} - 2.737648762 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right)} \]

      2. +-commutative 95.2%

         \[\leadsto 1 + \left(1.737648763 \cdot \frac{1}{\color{blue}{1 + 0.3275911 \cdot \left|x\right|}} - 2.737648762 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right) \]

      3. +-commutative 95.2%

         \[\leadsto 1 + \left(1.737648763 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} - 2.737648762 \cdot \frac{1}{\color{blue}{1 + 0.3275911 \cdot \left|x\right|}}\right) \]

      4. distribute-rgt-out-- 95.2%

         \[\leadsto 1 + \color{blue}{\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.737648763 - 2.737648762\right)} \]

      5. +-commutative 95.2%

         \[\leadsto 1 + \frac{1}{\color{blue}{0.3275911 \cdot \left|x\right| + 1}} \cdot \left(1.737648763 - 2.737648762\right) \]

      6. fma-def 95.2%

         \[\leadsto 1 + \frac{1}{\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}} \cdot \left(1.737648763 - 2.737648762\right) \]

      7. unpow1 95.2%

         \[\leadsto 1 + \frac{1}{\mathsf{fma}\left(0.3275911, \left|\color{blue}{{x}^{1}}\right|, 1\right)} \cdot \left(1.737648763 - 2.737648762\right) \]

      8. sqr-pow 53.4%

         \[\leadsto 1 + \frac{1}{\mathsf{fma}\left(0.3275911, \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|, 1\right)} \cdot \left(1.737648763 - 2.737648762\right) \]

      9. fabs-sqr 53.4%

         \[\leadsto 1 + \frac{1}{\mathsf{fma}\left(0.3275911, \color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}, 1\right)} \cdot \left(1.737648763 - 2.737648762\right) \]

      10. sqr-pow 95.0%

          \[\leadsto 1 + \frac{1}{\mathsf{fma}\left(0.3275911, \color{blue}{{x}^{1}}, 1\right)} \cdot \left(1.737648763 - 2.737648762\right) \]

      11. unpow1 95.0%

          \[\leadsto 1 + \frac{1}{\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right)} \cdot \left(1.737648763 - 2.737648762\right) \]

      12. metadata-eval 95.0%

          \[\leadsto 1 + \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot \color{blue}{-0.999999999} \]

   10. Simplified 95.0%

       \[\leadsto \color{blue}{1 + \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot -0.999999999} \]

   11. Taylor expanded in x around inf 98.2%

       \[\leadsto \color{blue}{1} \]
3. Recombined 2 regimes into one program.

4. Final simplification 97.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left|x\right| \leq 10^{-5}:\\ \;\;\;\;10^{-9}\\ \mathbf{else}:\\ \;\;\;\;1\\ \end{array} \]

Alternative 16: 53.7% accurate, 856.0× speedup

Math

\[\begin{array}{l} \\ 10^{-9} \end{array} \]

FPCore

(FPCore (x) :precision binary64 1e-9)

C

double code(double x) {
	return 1e-9;
}

Fortran

real(8) function code(x)
    real(8), intent (in) :: x
    code = 1d-9
end function

Java

public static double code(double x) {
	return 1e-9;
}

Python

def code(x):
	return 1e-9

Julia

function code(x)
	return 1e-9
end

MATLAB

function tmp = code(x)
	tmp = 1e-9;
end

Wolfram

code[x_] := 1e-9

Derivation

1. Initial program 79.5%

   \[1 - \left(\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(0.254829592 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-0.284496736 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.421413741 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(-1.453152027 + \frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot 1.061405429\right)\right)\right)\right)\right) \cdot e^{-\left|x\right| \cdot \left|x\right|} \]

3. Simplified 79.5%

   \[\leadsto \color{blue}{1 - \frac{0.254829592 + \frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}}} \]

5. Applied egg-rr 79.5%

   \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\left(\frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot -0.284496736 + \frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]
6. Step-by-step derivation

   1. distribute-lft-in 79.5%

      \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\frac{1}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)} \cdot \left(-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]

   2. fma-def 79.5%

      \[\leadsto 1 - \frac{0.254829592 + \frac{1}{\color{blue}{0.3275911 \cdot \left|x\right| + 1}} \cdot \left(-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]

   3. associate-*l/ 79.5%

      \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\frac{1 \cdot \left(-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}\right)}{0.3275911 \cdot \left|x\right| + 1}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]

7. Simplified 78.4%

   \[\leadsto 1 - \frac{0.254829592 + \color{blue}{\frac{-0.284496736 + \frac{1.421413741 + \frac{-1.453152027 + \frac{1.061405429}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}{\mathsf{fma}\left(x, 0.3275911, 1\right)}}}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right) \cdot {\left(e^{x}\right)}^{x}} \]

8. Taylor expanded in x around 0 75.3%

   \[\leadsto \color{blue}{\left(1 + 1.737648763 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right) - 2.737648762 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}} \]
9. Step-by-step derivation

   1. associate--l+ 74.8%

      \[\leadsto \color{blue}{1 + \left(1.737648763 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1} - 2.737648762 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right)} \]

   2. +-commutative 74.8%

      \[\leadsto 1 + \left(1.737648763 \cdot \frac{1}{\color{blue}{1 + 0.3275911 \cdot \left|x\right|}} - 2.737648762 \cdot \frac{1}{0.3275911 \cdot \left|x\right| + 1}\right) \]

   3. +-commutative 74.8%

      \[\leadsto 1 + \left(1.737648763 \cdot \frac{1}{1 + 0.3275911 \cdot \left|x\right|} - 2.737648762 \cdot \frac{1}{\color{blue}{1 + 0.3275911 \cdot \left|x\right|}}\right) \]

   4. distribute-rgt-out-- 74.8%

      \[\leadsto 1 + \color{blue}{\frac{1}{1 + 0.3275911 \cdot \left|x\right|} \cdot \left(1.737648763 - 2.737648762\right)} \]

   5. +-commutative 74.8%

      \[\leadsto 1 + \frac{1}{\color{blue}{0.3275911 \cdot \left|x\right| + 1}} \cdot \left(1.737648763 - 2.737648762\right) \]

   6. fma-def 74.8%

      \[\leadsto 1 + \frac{1}{\color{blue}{\mathsf{fma}\left(0.3275911, \left|x\right|, 1\right)}} \cdot \left(1.737648763 - 2.737648762\right) \]

   7. unpow1 74.8%

      \[\leadsto 1 + \frac{1}{\mathsf{fma}\left(0.3275911, \left|\color{blue}{{x}^{1}}\right|, 1\right)} \cdot \left(1.737648763 - 2.737648762\right) \]

   8. sqr-pow 40.8%

      \[\leadsto 1 + \frac{1}{\mathsf{fma}\left(0.3275911, \left|\color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}\right|, 1\right)} \cdot \left(1.737648763 - 2.737648762\right) \]

   9. fabs-sqr 40.8%

      \[\leadsto 1 + \frac{1}{\mathsf{fma}\left(0.3275911, \color{blue}{{x}^{\left(\frac{1}{2}\right)} \cdot {x}^{\left(\frac{1}{2}\right)}}, 1\right)} \cdot \left(1.737648763 - 2.737648762\right) \]

   10. sqr-pow 74.6%

       \[\leadsto 1 + \frac{1}{\mathsf{fma}\left(0.3275911, \color{blue}{{x}^{1}}, 1\right)} \cdot \left(1.737648763 - 2.737648762\right) \]

   11. unpow1 74.6%

       \[\leadsto 1 + \frac{1}{\mathsf{fma}\left(0.3275911, \color{blue}{x}, 1\right)} \cdot \left(1.737648763 - 2.737648762\right) \]

   12. metadata-eval 76.3%

       \[\leadsto 1 + \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot \color{blue}{-0.999999999} \]

10. Simplified 76.3%

    \[\leadsto \color{blue}{1 + \frac{1}{\mathsf{fma}\left(0.3275911, x, 1\right)} \cdot -0.999999999} \]

11. Taylor expanded in x around 0 52.4%

    \[\leadsto \color{blue}{10^{-9}} \]

12. Final simplification 52.4%

    \[\leadsto 10^{-9} \]

Reproduce

herbie shell --seed 2023196 
(FPCore (x)
  :name "Jmat.Real.erf"
  :precision binary64
  (- 1.0 (* (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ 0.254829592 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ -0.284496736 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ 1.421413741 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) (+ -1.453152027 (* (/ 1.0 (+ 1.0 (* 0.3275911 (fabs x)))) 1.061405429))))))))) (exp (- (* (fabs x) (fabs x)))))))